I would not recommend the idea you speak of unless you have a spare million dollars, and if you do, why not just get a brand new Lamborghini? Parts for exotic supercars are incredibly expensive, especially if you get one they don't make anymore. I've seen Italian supercars up for sale online and in car trader magazines for as little as $20000 but you don't want one. With a car like that, if it even runs it would be nothing but a headache and restoring it would cost you a small fortune.
A long time ago, on the original UK version of Top Gear, the hosts were given a challenge to find supercars with a budget of just £10000 (about $15-20k US) you may be planning on spending a little more but it doesn't get much better. Anyway, the quote in my signature comes from that film. All of those problems were occurring in a Lamborghini Uraco. One of those in good shape would be a beautiful thing, but it would cost you. On a budget it would probably give you more problems than pleasure. To really get a sense of what I'm talking about I'll give you the link to the cheap supercar challenge. First, you'll laugh your ass off, then you may reconsider this cheap exotic supercar idea. My suggestion: look for a Corvette instead. They're crazy fast, the parts are less expensive, and believe it or not, they're more reliable.
Anyway, here's that link. It's divided into 4 short videos, I would recommend watching the whole thing:
https://m.youtube.com/playlist?list=PL168DCC3AFDE3C046