Reverse gear making noise when car is cold - 1998 Volkswagen Jetta

if manual ( not specified) reverse gear is a spur cut gear and will be noisy when engaged
gets worse if the gear has been engaged inadvertently or slashed by mistake as small pieces of the teeth get chipped off and the result is a noisy reverse operation
nothing to worry about til it starts to clunk as you are going in reverse

reverse gear is what is called a spur cut gear and because of that it generally is noisier that the other gears
it runs on it's own idler shaft and there most probably will be a bearing problem that is allowing the mesh of the gears to alter and that will make the reverse noise louder than normal
so it is either a bearing on the way out or because of the way the gears are cut , a tooth could be chipped or partially broken off (slashing reverse inadvertently when selecting from 5th back to 4th)
the fix will be a box overhaul of the bearing or gears

Check for worn front drive axle CV Joints

Also check of worn motor & transaxle mounts

check the linkages from gear stick to gear box may be loose.

Motor or transmission mount/s may be bad .

From the manual:
The automatic transmission dipstick has a yellow handle.
Before checking the fluid level:

* The car must be parked on level ground
* The engine must be idling
* The gear selector must be in Neutral or Park
* The parking brake must be set

Reverse is not "synchronized" forward gears are, so it does make some noise, try shifting to 2nd before engaging reverse to slow gear rotation and limit gear clash or grind.

Clutch adjustment.

It could possibly be the joints in the exhaust manifold. The body and engine may flex differently in reverse. Since the engine is cold as are the pipes, a gap in a exhaust flange gasket could transmit a noise until everything heats up.

I would suggest hat you replace the timing belt. The noise that you hear is the belt.