It is intended to rise like a phoenix of hope from the ashes and misery of Haiti's capital, Port-au-Prince. An embodiment of the Haitian people's decency and courage – part resignation, part defiance – in the wake of tribulation and calamity, apparently without end. The two towers and steel-girdered roofing of the Iron Market – the heart and pulse of Port-au-Prince – have risen again and will be opened this week, on 11 January.

The date is a haunted and poignant one: the eve of the anniversary of last year's earthquake, which buried and killed 200,000 people, and has left millions homeless and bereaved. And the suffering goes on – Haiti is currently suffering a cholera epidemic, which a medical report commissioned by the French government has now confirmed to have originated in a river next to a base housing the very UN troops mobilised to help them.

"It's a celebration, but it's probably the saddest day in the history of Haiti," says Denis O'Brien, the Irish billionaire tycoon who has funded the reconstruction of the Iron Market. "We'll have to get the balance right."

In the context of devastation and desperation, the re-erection of the Marché en Fer has taken on a mighty significance for the country. It is this which has attracted the film director Patrick Forbes to make a documentary about the project, from scratch to the opening on Tuesday. And Forbes's central character is the ebullient O'Brien, the man whose outrageous mission has taken on such symbolism. "He's a bulldozer of a man," says Forbes. "George Howard, the site manager, calls O'Brien a ball-breaker whom you thank for breaking your balls."

The Iron Market reconstruction is the only visible sign of building in Port-au-Prince, and its cost – $12m – constitutes a tiny fraction of the $500bn of aid promised to Haiti, of which only $6bn has ever materialised. "Any sign of hope," says Forbes, "has taken on enormous significance." But its opening has already been postponed by a month by a combination of delays, riots that followed a contested election result and the closure of the airport. Even next week's opening is on a knife edge. "I think they'll make it," says Forbes. "Just."

Throughout its history, Haiti has suffered cycle after cycle of oppression and manipulation, political betrayal and tragedy, from the days of slavery under French colonial rule to the 2004 coup that unseated its first democratically elected president. But even by the standards of an unforgiving universe, the earthquake of 2010 was pitiless. No one who watched television, let alone anyone there in the aftermath, will ever forget the unearthing of dead babies and infants. The city of Port-au-Prince was literally levelled and hundreds of thousands were buried alive. Into the wreckage, along with the vital emergency work of the NGOs and the rock-star charisma of Bill Clinton, came missionaries who kidnapped children for "adoption" and vainglorious "philanthropists" who left once they had satisfied their own consciences but achieved little or nothing. Of the billions of dollars of aid promised by governments and individuals alike, little has arrived – either because it was offered out of vanity or for political gain, or because the crippling corruption of the Haitian elite would rather filter the money before it reaches its subjects.

The reason the Iron Market's success is so imperative, says Forbes, is that "it is the only reconstruction anyone is doing. When I first went to Haiti, I thought: 'Where are all the JCBs? Where's the rebuilding?' There was none. Only O'Brien's project. Apart from that, just all these guys clearing rocks."

Personally, I remember the Iron Market well, from several occasions buying voodoo ironwork I treasure. It is the kernel of the Haitian capital, a teeming, throbbing abundance of goods arranged along stalls of concrete, wood and cinder block; a multitude of people and kaleidoscope of colours – throngs of citizens come to gouge out a bargain, to equip their humble homes and kitchens with the bare essentials and maybe even indulge in a small luxury: pirate perfume, knock-off CDs, or a mobile-phone cover. This last is of especial cogency as the market reopens, for O'Brien made his fortune as founder and CEO of Digicel, which two-thirds of Haitians who can afford to own a mobile phone pay to chatter on.

The Iron Market, like almost everything else in town, was all but levelled by the earthquake. Its roofs and walls crashed to earth, though its two towers remained, albeit unsafe and in need of complete re-fortification. They are strange beasts: minarets built in Paris during the 1890s and originally intended for a station in Cairo but bought by Haitian president Florvil Hyppolite when the Egyptian deal fell through. The clock on the great facade between the towers stopped when the earthquake struck, and its rubble burned for weeks afterwards.

While the rest of Port-au-Prince toppled, one building had stood firm – pristine – among the wreckage: the Haitian headquarters of Digicel. In mid-January, O'Brien stepped forward to donate $5m to the Haiti Relief charity formed in response to the disaster. At the beginning of February, the mayor of Port-au-Prince named O'Brien as the stricken city's "Goodwill Ambassador" to the world. In addition to O'Brien's donation, a further $800,000 was raised for Haiti by a "text and donate" appeal to Digicel's mobile-phone customers in other parts of the Caribbean and Central America, where much of O'Brien's fortune has been made (the rest is from the South Pacific). Before long, O'Brien had become one of the key point men in Haiti for the Bill Clinton Foundation and Clinton Global Initiative, which – with Clinton nominated as UN special envoy to Haiti – were among the main propellers of the international effort to help Haiti.

O'Brien, whose residency recently moved from Portugal to tax-generous Malta, was estimated by Forbes magazine in 2009 to be worth $2.2bn (a speck of which he contributes to the salary of Ireland's football manager, Giovanni Trapattoni). He made his initial fortune with the sale in 2001 of his Irish mobile-phone network, Esat. Moving into the developing world, he then established Digicel in Jamaica. Before Digicel arrived in Haiti in 2006, only 5% of the population used mobile phones; today 30% own one. O'Brien has become the single largest investor in the country, with a total investment of more than $300m over the past four years. His company now has more than 2 million customers in the Western hemisphere's poorest country.

In keeping with the scale of his ambition, O'Brien announced his intention not only to rebuild the Iron Market but to do it to a timetable that seemed insane: within a year of the earthquake, in time for a grand opening on 12 December. The audacity of O'Brien's scheme attracted the attention of John McAslan, a Glaswegian architect whose firm, McAslan + Partners, has a special emphasis on social and community building; its projects have included the renovation of the Roundhouse in London and the award-winning ecofriendly headquarters of MaxMara in Milan. McAslan already moved in Clinton Foundation circles and had been in discussion about rebuilding the northern section of the market, which was destroyed during food riots in 2008.

"By mid-February," recalls McAslan, "we had a rag-bag collection of people in various places who were interested and determined to get it done within a year." The sense of urgency was not bombast on O'Brien's part, says McAslan, but "the fact that this was the only major reconstruction in Haiti, and therefore a symbol of hope. We had to proceed against immeasurable odds to keep the project on track."

Forbes' film, From Haiti's Ashes, opens with Bill Clinton promising Haiti: "You will not be forsaken, you will not be forgotten. America stands with you – the world stands with you." Clinton's rhetoric is not entirely without credentials: as president of the United States, he secured the return of Haiti's elected but deposed President Jean-Bertrand Aristide, toppling the savage dictatorship of General Raoul Cédras. But the memory of the political disappointment and ultimately turmoil that followed those euphoric days warns one against too much optimism. It is with some pain that one hears the voiceover that opens Forbes's film: "O'Brien and McAslan are Haiti's only hope of long-term salvation." Because the documentary which follows is about nothing if not the replacement and possible misplacement of Haiti's political hopes with a similarly wholehearted trust in Digicel, O'Brien's mobile-phone company.

The film is a suspense drama not about Haiti, but about O'Brien's project. An extraordinary team of people is assembled and battles to get the Iron Market built on schedule. O'Brien sets out his philosophy that "there is no point in being a conquistador capitalist", someone who echoes the smash-and-grab Spanish invaders of yore. "The conquistador was a robber," says O'Brien. "I hope that when I die no one will say I was a robber." He insists, for instance, that the original stallholders are invited back into the market when it is reopened, rather than "new people who are friends of friends".

O'Brien claims to love Haiti better than the Haitians who govern their fellow countrymen. "Even if a lot of the Haitian elite don't believe in their country," he told Time magazine, "I do." The country is run, says O'Brien, "by a handful of families" who have "no intention of releasing their grip". But O'Brien does not appear to have much intention of encouraging them to do so, even if, to cite Forbes's commentary, this really would be Haiti's "only hope of long-term salvation". No, O'Brien the benevolent magnate prefers to "get around" and "ignore" the elite, while McAslan the architect and pragmatist elects to work with it in the interests of the deadline.

The plan must be executed at "speed, speed, speed", insists McAslan, although the motivation for the rush seems to be O'Brien's deadline rather than Haiti's. When Forbes meets O'Brien in New York in December, he hears that a bus carrying steel workers brought in from Delaware has been attacked; O'Brien insists that crews will from now on be flown in aboard his private planes. However the true hero of the Iron Market saga, as portrayed in Forbes's film, is a nuclear engineer from Florida who is chosen to manage the site. George Howard marshals the forces, works with O'Brien to get steel roofing in from the port where it is stuck by red tape, contracts malaria and even cholera, and as the reconstruction slips away from its deadline, he has the nerve to admit that "this is certainly ego driven now" – only it is not clear whose ego he refers to.

All around the site and up the streets are plastic gazebos with Digicel logos emblazoned on them. For want of the market, the Digicel gazebos have become street stalls themselves. The people in one of the most deeply spiritual places on earth speak their faith among the crowds: "We say: Digicel, we love you!" cries one man. "We want to see you higher and higher."

When O'Brien makes his appearances on site, an off-camera voice says: "It's like a royal visit, isn't it?" Workers give him a globe, the world as a toy to spin. The most extraordinary scene in the whole film is a walkabout in the streets by O'Brien, during which a woman cries, as if in a voodoo trance: "Digicel m'a passé! Digicel m'a passé!" – Digicel passed me – as the CEO walks by her, invoking the healing shadow of St Peter. "Merci! Merci!" they call out as O'Brien examines a live turkey for sale. "Digicel for president!"

At a time when figures like Microsoft's Bill Gates and the Hungarian entrepreneur George Soros are deploying slices of their fortunes towards international aid and global health, O'Brien's Iron Market project is Haiti's miniature emblem of the privatisation of humanitarian policy away from national government and so-called "parastatal" organisations such as the United Nations. Neither Gates nor Soros has quite the direct and localised commercial interests that O'Brien has in Haiti – they operate to a different depth, scale and timetable – but Forbes insists O'Brien is part of the same phenomenon of "postmodern Rockefellers – in the same vein, only post-industrial". He refers to them as 21st-century "barons".

"This is the new reality," argues Forbes. "Of course the barons are not going to replace the parastatals, but they will change the landscape. This is about globalisation, it's about capitalism – a kind of parable of the modern world. You make a ton of cash, and it becomes less of a sin that you used that cash to make even more cash in a poor corner of the world if you use some of it along the way to do some good."

O'Brien's is both a success story and a vanity project. What we do not hear (or hear only privately) are the betrayals: the smaller "barons" who came to Haiti to establish charities of which they tire, who make promises they break, who cast out those to whom they gave a roof only for a while. For every baron like O'Brien, there are many minor aristocrats who leave a trail of disenchantment.

From Haiti's Ashes will be shown on the BBC