First, To be honest I don't know anything about the ae82s, but I have a issue with it. It's a 1988 corolla FX with the 4ac/4alc? Engine
Car drives good but in 4th gear, at anything over approx. 10% throttle , it pops out of gear and also in 5th if you don't keep your foot on the throttle and completely let off, it slips out of gear. My guess would be synchros, so please tell me if I'm right or wrong.

Also I have a FX16 transmission, is it the same as the standard FX even though the different engines?

Any help would help.

From memory there is a different final gear in the FX16 version but it would otherwise bolt onto your 4A engine without any problem.

yup the synchros are going bad. you have 3 options. 1st drain you tranny add some synchromesh ( autozone has some or GM dealers carry an effective one as well) and refill with redline gear oil or equivalent. this should solve or ease your gear popping. 2nd upgrade to a c52 tranny. any fx16 gts, ae92 gts, or 87-88 mr2 carry them. or even the c56 jdm tranny found in ae101 and ae111. all of these trannys are more stronger and less prone to gear popping out. 3rd rebuild you old tranny.

good luck
hope this helps out

Old post, but for reference, its probably not the synchros causing this, the infamous 5th gear pop. Now the synchros are likely worn out at the same time if your having this problem, but its not whats causing it. Its usually the bearings worn out with too much slop. The proper fix is a rebuild. Or find a replacement. MR2 has same transaxle, but the shift shaft comes out wrong side so you can't just direct swap it, you need to swap some parts.

sometimes shifter linkage bushing can cause this problems adjusting this is good place to start