Stick some pictures in with your next post showing what you've got to work with, how much of the shell is there? What engine you've got etc.
To answer your question properly we need a little more info but there is a plethora of information available for this kinda thing if you do a little searching..
What later model corolla did the engine come out of, what's the engine code?
You will need to use the ae transmission as the newer corollas are fwd and the 86 is rwd, specifically, a t56 gearbox. You can also use 'w' series gearboxes with minimal modifications.
Cost varies, how much can you do yourself, how much will need to be done by a garage. The more you can do yourself the cheaper it becomes. I would say that most people can do all the mechanical side of things with some patience and time spent researching the topic. I would however, recommend getting the wiring done professionally, save yourself the headache.
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