From outside, hard to say what gearbox you have, have a look at the casting year on the housings and also if it is a mechanical speedo sensor or an electronic one. It may help to determine what could it be...but if it has been dismantled, you cannot be sure that what is inside is in accordance with the outside...
Also you can rotate the input shaft by hand and calculate how many degrees it makes the shafts to turn. The ratio can give you accurate hints of what is inside.
Asked for 4th as I already saw a rebuild with a "builder" who forgot to replace the locking ball to the link that handles the fork, so obviously the gears didn't stay engaged. In the UN, 3rd and 4th are held by the same link so if the locking ball was missing the problem would occur on these 2. As it only happens on one, forget this way.
Neverthless as a jumping gear has many sources and one of them is the linkage, check that when you engage manually the gear using the arm on the rear side of the gearbox, you fill it "locked" when in 3rd gear. Too much play can drive to this trouble. Also check that your NSX wires allow the gear to be fully engaged.
If all is ok from this side...the problem is eventually inside : could be the 3/4 hub, hub spline, dog teeth of the 3rd pinion, fork damaged, synchro ring or just the 3 springs that are worn...For sure, one of these..., but unfortunately only a visual inspection will tell you....
Cheers
Stephane