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On a $150 car, I'd do a compression test and expect to find a bad head gasket, also check for rust in the structural areas above the rear wheel arches, be aware that the undercoating can conceal major problems.
Main thing you need to be aware of, as a n00b to the MR2, is it's cooling system. Unlike most other cars, a bit of air in the coolant lines is a big problem, can cause overheating. If it overheats, it can blow out the head gasket. Get the BGB, follow proper procedure for getting the air out of the system. For this kind of race--I'd install all new coolant hoses, if you blow your cooling system the race is over. Definitely test drive a few hundred miles ahead of time.
On the 86, 5th gear pop-out is a real common problem. Of course, you can go plenty fast in 4, and you can always drive with your hand on the gearshift.
Best mod for power would be to add a supercharger, you might manage on your budget if you're good at scrounging.
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