Only really problems encountered with covering such high miles, apart from a cylinder head gasket at 48,000 miles and again at 90,000 miles.
Replaced the alternator at 102,000 miles, seems to be a common Punto problem, but mine lasted longer than most.
Front and rear suspension totally replaced at 100,000 miles due to wear, components bought secondhand off a 1,000 mile smashed Punto Sporting, now handles like a dream, with very little body roll, although ride is slightly harder than before.
Great car, a breaze for DIY, dealer service once - really unimpressed, carried out all servicing work for last 80,000 miles myself including the head gasket and timing belt changes, reasonably simple jobs due to small engine in large engine bay.
Even with 131,500 miles the engine still pulls strongly, in fact it gets better with age, goes 20mph faster than the manufacturer's top speed, without a hill or any wind!!
Not a Fiat of old, I have never had an electrical problem yet, and no rust. Not even failed an MOT yet.
Really comfortable car with height and lumbar adjustment on the seat and height adjustment on the steering wheel. Even with miles the interior is as good as new.
It may only have a 1.1 engine but most people think it has a 1.3 or 1.4 engine mainly due to the pickup speed and the fact it doesn't slow with a full load.
Downsides - front brake pads wear quite quick, and a bit noisy above 70mph due to high revs, needs at least one more gear.
A real roomy small car with plenty of get up and go and well equipped in SX trim anyway.