I'm at Hakkasan in Mayfair, in a state of extreme anxiety. It's not only that – girl crush! – Debbie Harry was the first pop star I ever liked (I was nine when Blondie's Denis reached No 2 in the UK chart, and I remember watching her on Top of the Pops like it was yesterday). Nor is it that, hearing her on Desert Island Discs the other morning, she sounded just a touch tricky. No, what it really comes down to is this: while she continues, at the grand old age of 65, to be one of the coolest women on the planet, I remain, at the grand old age of 41, one of the squarest. How did this happen? Why did I not follow her example? And how much will she notice that I didn't?

She arrives about 20 minutes late, which really doesn't help: a vision in black Lycra (or something else quite stretchy) and big sunglasses. But I like her immediately. She is kind. She exudes good manners and a certain kind of professionalism, and takes it in her stride when, even as she is still perusing her menu, I blurt out what must surely be the daftest question she has ever been asked. So, I say, what did you eat when you were a punk? Truly, it's hard to tell which of us is the most startled by this. "What did I eat when I was a punk?" The briefest pause. "Well… practically nothing. We were broke. We used to go to Chris's mother's house [Chris Stein, Blondie's guitarist and her ex-boyfriend] and have this kosher beef dish she made with lots of garlic in it. We went to Chinatown for dumplings. We used to have canned chilli. God, I don't even remember! I would make baked yams, which were healthy and economical. I can cook; not great, but I can. I like cooking for other people, but I don't for myself. I mean, why bother?"

We're sitting side by side at a tiny corner table, which means that I am able to examine Harry's face minutely. She looks amazing: peachy is the word. A lot better than she did 10 years ago. "I was in a slumpy kind of period then, and I'm not now. I'm on a big regime because I want to feel my best. I have a trainer. He's a terrible monster, but I love him." It's calisthenics, really, and it's hardcore. She had a facelift in the early 90s, a fact she has never tried to hide. "It's holding up well," she says, running her fingers lightly over her cheekbones. Was it frightening? I would be worried about looking like a Mekon. "Not really. I was careful about who I went with. I looked around, I interviewed. For me, it wasn't just a madness. That does happen. Panic sets in, and people make bad decisions. But not me. I did my homework. There were places I went to that were really scary, and I ran out."

Would she still have had a facelift if she hadn't been a pop star? "Yeah. I think if I'd been very depressed about myself, I might not have done. But it gave me a certain encouragement and satisfaction." Of course ageing makes her anxious. Am I kidding? "It's the same for everyone, isn't it? Everyone, at some point, has a realisation about their longevity, or lack of. How you respond depends on how much you like living. I think for men, the ageing process is more evenly paced. Women have their ups and downs: all of a sudden, a page is turned. It really is radical and hormonal. Women just have a more rigorous time physically than men. It's a challenge. Sometimes I would love to eat a whole pizza. But then I follow it through in my head… You eat the pizza, and then you go home and feel rough about yourself. You're just going to regret it. I don't kid myself about that."

I don't tell Harry that I often eat a whole pizza. But perhaps she will work this out for herself. Because when we order, it's me that goes for the fatty stuff: abalone puffs, duck rolls. She limits herself to a tiny bowl of soup and a selection of delicate-looking dumplings, not all of which, it is patently clear, she intends to eat. This is not to say that she isn't enthusiastic. "That smells good, doesn't it?" she asks, dipping her head over her bowl. She tastes a little. "This is super-delicious! It's gorgeous. It's elegant."

Harry is in London on a recce ahead of some dates Blondie will play here this summer, when they promote their new album. Did she expect to be touring in her 60s? No, not for a minute. "I did those solo albums at the end of the Eighties, and they didn't do very well, and I had no inclination to do Blondie again [the band split up in the early Eighties and did not reform until 1997]. I like performing, so I probably would have figured out some kind of smaller act, a cabaret thing." She belongs, she readily admits, to the first generation of ageing rock and pop stars, though in her view 70 is the new 50. How will she know when to stop? "I guess when there's no longer an audience. But I've gone this far…" From deep inside her there comes a croaky kind of laugh.

I love it that Harry isn't a moaner. "Fame is important so far as having an audience goes, and enduring and stuff like that. How else would you do it? So you don't complain about it." But most famous people do. They complain about being followed – by their fans, by the paps. She shoots me a look. "Well, you can always tell them to fuck off. No, I appreciate my fans. How would I exist without them? Most of the time people are nice to me, and if I'm out without make-up and stuff, and not ready to be photographed, I just say: 'No.'" Wow. Quite sane. "Well, I suppose I have a bit more anonymity than Madonna, or Gaga. They're super up-there. I'm more of a cult figure." She thinks for a moment. "When I was younger and people were bothering me, I don't know if I enjoyed it all the time. But I sometimes did."

Her parents wanted her to marry and have a family, though this never happened; in the 80s she nursed Stein through a serious illness but then they split up, and she is now godmother to his children with someone else. Her parents only began to grasp the extent of her fame when people came into their gift shop in upstate New York and asked about her. The rock chick daughter was the last thing they expected. "But then, if I'd been my parents, I wouldn't have thought it would have happened to me either. I was very shy and quiet, and all my ambitions were dreams. I hadn't a clue about the music business: just this drive, this obsession, that I can't explain." Is it lonely, life on the road? I hope she has someone waiting for her in New York. "No, not particularly. There's no one pacing round the house. I date. I don't love dating, but I hope to do it more; I wouldn't say 'no' to one wonderful relationship." Do people fix her up? "Sure, and sometimes it works and sometimes it's horrible." Do men find her intimidating? "I suppose some do. But that's their problem, and ultimately it's not very interesting to me." Is she a feminist? "How can one be a woman and not be a feminist? That's my question."

I'm calmer now. Debbie, ever the pro, has created the soothing illusion that we are bonding; she even tells me that my questions are, well, if not exactly interesting, a nice change from all the music stuff. But then her PR appears and, in my panic – there is still food on the table! I was hoping to eat it! – I well and truly fluff it. As we wait for our waiter, Harry asks me if I like cooking. Yes, I say, very much. I am ridiculously pleased by her interest. So ridiculously pleased, in fact, that, without properly thinking about what it is that I'm about to do, I reach into my bag, pull out my phone, and proceed to show her a photograph I took of the salad I made last night. "Oh," says Harry in her light, rather distant voice. "Oh. That's… beautiful." She hands my mobile back to me. Then she turns to her PR. "Please make sure Rachel gets tickets to see us," she says. She smiles at me, beatifically, then she gathers herself and leaves.

Soon after, I do the same. Though with slightly less aplomb. Oh God. I can hardly believe it. I had lunch with Debbie Harry, and what did I do? I showed her a photograph of... a salad.

Blondie's The Panic of Girls is out now. They play Kew Gardens and Somerset House, London on 11 and 13 July, and festivals throughout the summer