When Josie Rourke, the artistic director of the Donmar Warehouse, first began directing professionally at the age of just 25, she gave herself a makeover, the better that she would be taken seriously. She cut her hair brutally short, wore glasses all the time, and was never seen out of trousers. "I'm sure I looked ridiculous," she says. "I was pretending to be someone I wasn't, and I had to have a word with myself about it. I think authority comes from authenticity."

These days, of course, she knows precisely who she is – which is to say that she is a woman who is as happy to discuss the loo situation at the Donmar (somewhat minimalist) as she is its next production (Conor McPherson's 1997 play The Weir, directed by Rourke herself and starring Brian Cox and Dervla Kirwan). In 2016, when the theatre building finally comes into the ownership of the Donmar – at present it is leased – she intends to lead a charge on both its lavatories and its bars, and she doesn't care who knows it. "It sounds shallow, but I'm completely obsessed with making sure that it has a great bar and a lot more loos, and I want to take this opportunity to apologise to the public for the loos at the Donmar…" She laughs. "In fact, maybe I should get rid of all the bars and install 87 loos instead."

It's now a little more than a year since Rourke, who is 36, stepped into the (pretty enormous) shoes of Michael Grandage at the Donmar: previously she ran the Bush Theatre, whose financial future she more or less single-handedly secured, and whose move to a shiny new home she oversaw. The pace is, she admits, ceaseless: the Donmar's small team stages six productions a year. But since this speed is something she craves, it bothers her not at all. "I love the energy of a deadline," she says. "I'm obsessed with movies about the news: The Front Page, His Girl Friday. I love the idea that you have to hit it completely, or you've messed up. Do I go home and stress? Well, that's one of the things that home is for, and I'm lucky that I've always slept well."

She points happily in the direction of a cardboard model in the corner of the room. Once The Weir has opened, she will turn her attention swiftly to The Machine, a new play about Garry Kasparov's controversial battle with Deep Blue, an IBM computer, in 1997. She will direct it for the Manchester International Festival, where it will be staged in the Campfield Market Hall; the model is a mock-up of it. "I'm from Salford, and I've never done a play in Manchester before," she says. "When I started out, I couldn't get arrested in Manchester."

So, why The Weir? Her instinct was that it would make for a good fit with the productions that preceded it: an exceedingly jolly revival of Pinero's theatre comedy Trelawny of the Wells, and Phyllida Lloyd's acclaimed, all-women Julius Caesar. "It's all about rhythm. They were both quite expansive pieces, with a very distinct style. The Weir is a piece of poetic realism, a piece of storytelling. It says: lean in, come closer."

The play is set in a rural Irish pub. "We had the 'stagger through' today – that's our hard-wired euphemism for the first time you run a whole play together." And what did she think? "I thought: wow, that's the work we need to do. It's suddenly very clear, like one of those magic eye pictures."

But she sounds so calm! "Yes. Sam Mendes [another former artistic director of the Donmar, whose assistant she was for a time] once told me: 'It's the job of the director to absorb stress, not to give it off.' And he was right. Sometimes, its necessary for the actors to get the jitters. They might need them to give themselves a kick up the arse, or to get themselves into the zone. But you're the calm person at the centre of that. No one wants to see a director panic."

What's it like, I wonder, bossing Brian Cox around? "I think directing is like acting: it can only come from your centre. It's about weighing up the room, understanding that different people have different paces, seeing what tension you can diffuse. It's about how you notice people, and how you give attention to them. And the more you do, the more confident you are."

Rourke trained at the Donmar back in 2000, on its assistant director scheme. "I was stupidly lucky. I had this incredible compression of experience, assisting Sam [Mendes], Michael [Grandage], Phyllida [Lloyd] and Nick [Hytner]. You couldn't have a bigger range of presences within the room than that. And then Sam offered me my first show, and I was away… It was crazy. I was a very young woman when I started to tell very established actors what they should be trying, and it was scary at first. The only real wisdom I have about it is that you need a preparedness to take yourself out of the situation. There is no room for your ego as a director. Your job is to get people to rub along with each other."

Has any actor ever told her, on hearing her notes, to get lost? "Yes, all the time!" So what happens then? "You have a responsibility to follow your actors' instincts. Someone might be offering something interesting, or good. But in the end, good directors never give up. It's about rigour. It's not about knowing the note. It's about knowing when to give the note."

Rourke, small and dark and ever-so-slightly given to speechifying, isn't from a theatre background; her mother was a teacher, and her father was an accountant (useful – she sometimes gets him to look over her budgets). "I was an extremely bookish child," she says. "But I was sociable, too. I finished reading Howard's End in the toilets of the Hacienda. I was that kind of kid." Her Catholic school didn't put on plays, and she thought for a long time that she would be an academic – "maybe the kind who wrote novels, too" – though she went "obsessively" to the theatre, queuing up for cheap tickets at the Royal Exchange in Manchester. "Then I swapped my school for Eccles Sixth Form college, and I was Olivia in Twelfth Night." This was when it happened: suddenly, there came a moment of great clarity. "There I was, giving my Olivia, and I thought: I'm awful, she's really bad, he's standing in the wrong place, and that light should not be green."

She read English at Cambridge. "That was a big deal; I was the first person in my family to go to university, and the first person in the history of my school to go to Oxbridge." There, she did "loads" of directing, and by the time she left, she knew she wanted to try and have a go at doing it professionally.

Her parents were incredibly supportive, for which she will always be grateful. "First, I went back up to Manchester, where I worked off my debts temping. I used to type up letters and then put them in this kind of leather menu card that said 'for your signature' on it in gold writing. Then I thought I'd better go to London. I got a job working nights as a banking secretary in mergers and acquisitions; that gave me the mornings to try and do theatre stuff."

Nine months later, she was taken on as a trainee at the Donmar. "I was incredibly gauche. I still don't know how I got the job. In the interview, Sam admired my jumper, and I said: 'Thank you. My mother gave me £30 for an ironing board cover, and this is what I spent it on.' "

Her path since has been smooth and swift, almost embarrassingly so, she says. But this isn't the only aspect of her career that is anomalous. Her sex also marks her out. Other women in positions as senior as hers remain a rarity; a fortnight ago Nicholas Hytner announced that he will step down from his role running the National Theatre in 2015, and the runners and riders to succeed him seem mostly to be men. (Marianne Elliott, the director of War Horse and an associate director at the National, has already ruled herself out.)

"We are feeling change," insists Rourke. "When I started directing, there weren't many women directors, but now there are some fantastic ones coming through: Lyndsey Turner, Polly Findlay, Carrie Cracknell." Yes, but they're not running institutions, are they? "Well, there are directors who don't want to do the bit of the job that involves fundraising, HR, management, programming, budgets, reporting to boards, reporting to the Arts Council and talking to journalists about the cultural landscape. They're very clear about that, and I think that's OK. There isn't a requirement to do it. But I do think we need to do more around the time when people are deciding to have children. These jobs are not well paid; our theatres are driven by passion, not by financial gain. However, at the tipping point when you are going to try and have a family and work in a theatre, that might be a factor, and we need to think about it."

Rourke sees herself very much as someone who has been the beneficiary of proper public funding for the arts – and this is her favourite argument when it comes to the matter of government cuts: that they will exclude the young from the theatre. "It's about making sure we don't lose a whole generation of audiences."

She has met Maria Miller and is hopeful that the new culture secretary, having spent some time listening to those who run our theatres, will soon clear the air. "It made the sector nervous, when the government talked about philanthropy at the same time as it talked about cuts. To encourage more philanthropy, we're going to need a big cultural shift, and that will take years, and it won't plug the gaps in the meantime. We need to be honest about that. We also need to acknowledge that it is much harder in regional theatre."

In general, though, she remains optimistic; the lights aren't about to go out just yet. Partly, this is a matter of personality. Partly, it is a matter of experience (the Bush survived, didn't it?). But mostly, it's just down to common sense. "The government would be completely bonkers to decimate theatre in this country," she says. "We make theatre like the Brazilians play football – and why you wouldn't want to preserve something that precious, I just don't know. "