Yes, heat can cause it to "thermal throttle" in which is slows down to maintain the heat level without having to shut down.
Cooler to buy, depends on how much you want to spend really. You can get a decent air cooler like the Evo 212 for about $20-$30, you can get a all-in-one water cooler starting around $50 or so, or go full out water cooling and spend $1000+.