Sounds to me like you could do with a new battery and a new starter motor, and have a mechanic give the car a once over just to be sure. Better to be safe than sorry if you ask me
Also, if you have just bought the car or not had it long, you should check that the cambelt/timing belt has been changed (they should be changed at around 60k miles).
Another thing to look out for are radiator switches, radiator fans & motors... I had to change mine as soon as I Bought the car as the fan was not kicking in and I didn't want the car to overheat...
Another is the City button (I'm not sure if your car will have this) but there are quite common problems with these.
Any problems with Punto's should be easy (and reasonably cheap) to fix, as long as they are not left to get worse. You will love the car and find hardly any problems, as long as you drive it right!
Good luck with the car and enjoy the modding & racing!