I'm not big on Damien Rice, and I think that the songs in this movie DO sound a bit like his, but I also think that the dude rocks a little harder. I like the moments when he's just singing by himself more than the duet ballads. Overall, though, I would say that the songs fit in perfectly with the movie, and I loved it as a whole. I don't think I would buy the soundtrack and listen to them, but they work well in the context of the film. Overall I thought it was a wonderful movie, and I've been recommending it to everybody. It's kind of feel good, even though I cried a lot.
Also, in reference to that Umbrellas of Cherbourg recommendation--that's one of my favorite movies ever. I'm not sure if that recommendation was in discouragement of Once, though.