Transmission or clutch

Sounds very much like a clutch problem, reverse has no syncro so it is always near impossible to select with a faulty clutch, i would get a mechanic to inspect this first hand to be sure and go get a few quotes as well don't settle for the first price you get, it pays to shop around.

How do you adjust the cable clutch for a 1996 VW Golf? Can you adjust it before you install or does it just self adjust once you install?

This should be a self adjusting clutch cable... however, in order
for the mechanism to work correctly, you need to first "collapse"
the cable so that it can be installed. Some do not have a cable fyi

Once the cable is collapsed (outer accordion shell compressed), attach
the cable at both ends and then it can be extended.

Pump the clutch pedal 5 or more times to allow the self adjusting
mechanism to find the correct adjustment.

In order for the cable to stay collapsed, it should have a plastic
tie-strap of some kind, that hooks into the lower part of the accordion
bellows, to keep the bellows collapsed... see image.

Thomas