kelemvor is definitely correct in that you don't need some expensive massive wattage power supply - most MAME machines are about as bare as it can get in terms of power requirements. That said, it should probably be stressed that this doesn't mean to simply buy the cheapest power supply you can find. It's worth spending at least a bit of money on a power supply that isn't bound to fail in 3 months, taking the rest of the computer down with it.
I'm partial to FSP for lower-end requirements due to good experience in the past, but there are quite a few good brands out there...Seasonic, Sparkle, Silverstone, PC Power & Cooling, and a bunch of others. Its a fairly sure bet that if its a good buy, there will be some reviews saying so - never hurts to put the model in google before buying if you're not sure.
Thermaltake isn't a brand I know too much about, but from what I've read and seen so far, they're putting out some rather good quality stuff.
Cooling is the other major concern I have in something like a MAME cabinet - most cabs tend to have the computer tucked away in some enclosed section that just traps heat, while frequently running 24/7. I personally favor just mounting the components directly inside the cabinet instead inside a case, inside the cabinet. A couple motherboard standoffs and well placed L-brackets for the power supply and hard drive works well at keeping everything in place for me. Airflow is key - make sure you have at least some air circulation to the computer, putting it in multiple layers of cases is going to make that harder to effectively do.
Another option would be to buy a motherboard tray (something like
this) - gives you plenty of open space, but you still have a rigid frame to ensure something like your Arcade VGA card stays securely in place. It also takes out the work of measuring and determining all the points to drill for mounting motherboard standoffs to your case.