From a certain perspective, Jesse Eisenberg's acting career resembles a darkly ironic cosmic joke, with Eisenberg as the victim. The story goes like this: you are a shy and awkward child, deeply uncomfortable in your skin. Then, one day, you discover acting, which you find "enormously comforting". Performing a prescribed role soothes your paralysing self-consciousness; hiding inside a character, you get the attention you crave, minus the deer-in-the-headlights panic of trying to "be yourself". Unsurprisingly, you prove particularly good at portraying fiercely intelligent but emotionally semi-detached geeks, angry at the self-assured world from which they, like you, feel excluded. You do it so well that eventually, at 27, you're cast in one of the highest-profile Hollywood movies of recent years, starring as Mark Zuckerberg, the founder of Facebook, for which you bag an Oscar nomination and justified critical acclaim. Which means that half your life, these days, is spent doing precisely what you'd fled into acting to avoid: being paraded around the movie-publicity circuit, not in character but as Jesse Eisenberg, put on the spot by talkshow hosts, stopped in the street by admiring fans and hassled by gaggles of journalists demanding that you tell them about yourself. No wonder you wear your hair in thick, curly clumps, almost down to your eyes: it's the only protection you've got left.

Not that Eisenberg is moaning about his situation. Arriving at a coffee shop near Grand Central station in New York, bike helmet in hand, he is on the contrary unfailingly courteous, posing patiently for photos with customers. ("Thank you, thanks for coming, thanks for doing this, thanks," he says to me as we sit down. This is supposed to be my line.) He is, of course, self-conscious about being so self-conscious, and so quickly grows uneasy talking about how much more he gets recognised in public since playing Zuckerberg: "Complaining about it is obnoxious, because it's the problems of the rich, you know, isn't it?" But it's clear he's deeply uneasy. "The thing I find most alarming," he concedes under questioning, "is that I've only been famous for three months, but it's been the three months [in which] anybody who didn't have a camera on their telephone before, now they do. So, in the subway, I'll turn around and see someone holding their cellphone up or their iPad and surreptitiously taking a picture of me, and it sort of feels... I mean, not to self-aggrandise, but it sort of feels like an attack. It's jarring." His eyes dart beneath his fringe. This is the final twist in the cosmic joke: Zuckerberg and Facebook may or may not be destroying our privacy, but The Social Network – which is in large part a meditation on that question – certainly seems to have ruined Jesse Eisenberg's.

It's a testament to Eisenberg's talent that so many of the characters he plays are of a piece, yet he never radiates the tedious Hollywood sense that he's just playing himself, or that he's a one-trick pony. His roles as the frustrated virgin Nick in 2002's Roger Dodger and as Walt, the furious and defensively contemptuous child of divorcing Brooklyn bohemians in 2005's The Squid And The Whale, seem in hindsight like logical forerunners of his portrayal of Zuckerberg, who channels similar resentments into billionairehood. (He also starred in two well-received 2009 comedies, Zombieland and Adventureland.) Eisenberg would probably sooner die than express the view that he ought to have walked away with Best Actor for his compelling performance in The Social Network, which manages to portray the Facebook founder as simultaneously intensely dislikable yet oddly heroic. But come on: in the absence of Oscar-bait period drama about a lovable British monarch, he surely would have done.

Following a brief detour to voice the part of a (geeky) pet macaw in the animated movie Rio, he can be seen next in the UK in Holy Rollers, directed by Kevin Asch, the true story of Hasidic Jews from Brooklyn lured into an Israeli ecstasy-smuggling ring in the late 1990s. Made before The Social Network, it's both less laugh-out-loud funny, and more engrossing, than its unpromising title suggests. Eisenberg plays Sam Gold, a 20-year-old whose naivety – at first, he thinks he's couriering medicine – makes him an easy target for smugglers who rightly predict that devout-looking men in black hats, long black coats and sidelocks are unlikely to be bothered by customs agents. The barriers that separate the youthful Hasidim from the New York that surrounds them aren't so different from those that separate Zuckerberg from Harvard's preppy elite. "It's different cosmetically, of course," Eisenberg says, "but it's still just a question of trying to bring some emotional realism to a character that's living in a very isolated world."

To research the role, he made multiple visits to Hasidic schools, and was given the barmitzvah that was absent from his secular Jewish upbringing. The seemingly isolated sect proved surprisingly welcoming. "They'd talk to me for hours and hours, and almost never question why I was there. One guy did ask, and I told him I was doing a role in a play – I thought if I said 'movie' it would lead to too many other questions – and he was, like, 'Oh. OK.' And that was it."

One might have assumed that someone as ill at ease in interviews as Eisenberg would clam up when asked to talk about himself; in fact, though, his dislike of such questions seems based more on the fact that he can't help but answer them candidly, in long, endlessly detouring flights of introspection. This isn't a matter of shtick; it's all real. Compliments seem genuinely to pain him ("The more people say nice things about me, the more I feel it's false," he told one interviewer). Being recognised in the street alarms him (though it has its upsides: "It makes you not want to do stupid movies, because you end up having to talk to people who like stupid movies," he has said). Success depresses him ("If you look at the movies that come out, most of them are bad, so it's not as if achieving some level of success means you get offered better roles, because frankly they don't seem to exist"); appearing on TV chatshows makes him paranoid ("I tend to feel I'm being mocked in some way"). He writes plays, with roles for himself, and there are plans to produce one of them in New York soon, but he worries that his writing might be rubbish and that nobody is willing to tell him.

Getting all this off his chest confers little relief, however, because then he just feels self-indulgent. When I point out that journalists, at any rate, prefer candid interviewees over hyper-controlled actors determined to give nothing away, he looks stricken. "But don't you think that's a good thing – saying less? Don't you think it's healthier than, you know, constantly revealing your deep fears? I mean, there's something almost more obnoxious in what I'm doing, isn't there, as though the world needs to hear all about my Jungian theories of myself? Right? I mean, wouldn't it be more respectful for me to say, 'This movie's so great! We're so excited that it's coming out in England'? Wouldn't it?"

Close family ties – plus the services of two therapists – are what seem to keep Eisenberg from drifting unrescuably into a vortex of anxiety and self-doubt. He was born in Queens, and his parents still live in New York; his younger sister Hallie, also an actor, appears in Holy Rollers. When we meet, he's en route to his weekly visit to his 99-year-old aunt, who lives uptown. His father teaches sociology in Brooklyn, and, unlike when he was a teenager, Eisenberg is now unembarrassed to discuss the fact that for two decades his mother worked as a children's party clown. "She didn't do my parties," he clarifies hastily. "That would have been strange. What she would do would be to barter with a local magician, so she'd do the magician's kids' parties for free. Although the magician cost a little more, so we had to give him a pie as well. To make it an equal transaction.

"Anyway, as you might imagine, she's very supportive of the arts. A lot of the things I do are so ridiculous, but not at all compared with what she's done. So that's nice... she knows what it's like to put yourself out there and make yourself vulnerable in order to entertain."

Eisenberg now lives in Chelsea with his girlfriend Anna – a 33-year-old executive at an arts education organisation who surpasses even him in publicity-shyness – in what he describes as "a very small apartment with no internal doors". They share it with a changing selection of troubled foster cats. (Eisenberg once said, presumably not entirely in jest, that he fosters new cats in proportion to how guilty or self-loathing he feels about each new professional success.) "Shanti has a neurological disorder," he says, describing the current residents, "and Snow Leopard has some kind of childhood trauma where he doesn't leave the closet. But they're so lovely. My job is a strange one, with no routine, so it's nice to have some kind of responsibility, even if it just means filling up two bowls."

Given Eisenberg's neurotic candour, it comes as something less than an enormous shock to learn that one of his next roles – after the crime caper 30 Minutes Or Less, due for US release this summer – is rumoured to be in Woody Allen's next movie, alongside Ellen Page, Alec Baldwin and Penélope Cruz. "The way he makes movies is so secretive, I'm not sure what the official status of that is," Eisenberg says. "But, yes, I am aware of it." Speaking of Page, the star of Juno, it turns out there is at least one identifiable upside to Eisenberg's new-found profile: he doesn't get mistaken for Michael Cera so much as he once did. "I guess, you know, I was in something that became part of the popular consciousness. So now," he adds, gnomically, "people mistake me for myself."

He is, he insists, less anxious than he once was. "I know some amazing actors who are not mortified every moment of the day," he says, hopefully, "so my feeling is that maybe you don't have to be a wreck to be good." But does he ever worry that successful therapy, or just growing older, might cause his anxieties to subside to such a point that he's no longer a virtuoso of on-screen awkwardness, too serene and comfortable with himself to portray discomfort and isolation so well? A flicker of a grin: "My therapist says I'm never going to get to that point, so I don't think it's going to be a problem."

And shortly thereafter he's back on the street, helmet strapped on, and launching himself off into the 42nd Street traffic, weaving perilously between cars and taxis, taking facefuls of exhaust fumes from buses – just another anonymous Manhattan cyclist, which is almost certainly exactly the way he prefers it.

• Holy Rollers will be in UK cinemas on 15 July