I have an '05 and I am a bit surprised at the amount of surface rust under mine. A lot of the suspension components are really quite rusty and bolts would take a bit of shifting. If I compare it to my 12yr old Volvo, the Volvo is in much better condition underneath, with barely any rust at all. Also the paint on the Jazz is not as good - chips easily and subsequently rusts. I have to say it has the worst paint of any car I've recently owned. Not saying its bad, but European paints are better, inc my old Citroen. I'm going to get the paint chips / door edges repaired professionally this summer to keep it at bay.
I do think the previous owner had only used car-washes as the underside was caked with old dirt. You can't beat a hand wash with hosing the sills/arches every week, esp in winter. My Aunt has an '03 Jazz and I checked under hers and hers has much less rust, but where she live the roads aren't salted.
If it's clean underneath then I expect you can keep it that way. I suspect rust will be the eventual killer of Jazzes, I predict many will succumb fatally by 10 yrs old.
No other probs except a sticking brake.
I'd be interested to know if others have noticed rust under theirs. Just stick your head under the back bumper and check the back beam axle. The other place us the main front suspension strut (what the coil spring goes round). I can flake of rust with my fingers and the nuts have seized on the bolts. They would have to be cut out if they ever needed shifting.
Other than that, it's a great car, Honda make great engines and the Jazz engine is no exception. Since engines are the most complicated bit of a car, and therefore the most expensive bit to fix, this remains a strong virtue of the Jazz.