Krem's about on target... the hard part isn't moving the 5 speed to the car with the auto, but moving the Auto to the car with the 5 speed will most likely suck. I don't think the 5 speed has a chassis harness for the automatic ECU (I know mine doesn't, so, you can probably add pulling the dash, center console, steering wheel out of BOTH cars... check ALL of the sensor and control locations before beginning work on this project. Along with swapping the radiator (piece of cake), you'll need to swap over the hard pipes for the transmission cooler too.
I'm not certain if the Automatic tranny wiring is in a separate harness or if it was a different chassis harness altogether, so check and double check before starting.
You DO have to swap the ECU's as well, so, if your other Supra is newer than '88 you're kind of screwed... unless you feel like doing some hacking to both chasis harnesses.
Have fun.
Personally, I'd sell both and find a nice 5 speed turbo MKIII... they almost all came with the sport roof.