Will my fridge be damaged if I put a bag of ice in the fridge compartment while I wait for the repairman to fix a warm fridge/cold freezer?
It depends on the fridge, but I doubt it would cause additional problems for the unit.

I wouldn't trust it to keep your food safe, though.  A single bag of ice is cold, but won't cool a large space.  If the air circulation in your refrigeration compartment is broken, the food more than a foot away from the ice is unlikely to be cooled much.  If the air circulation is operating (but blowing warm air), it could overwhelm the cooling effect of the ice.

I'd try to put everything necessary into chest coolers (maybe ask a neighbor for an extra one if necessary), then ice or dry ice from the store to keep it cool overnight until the repairman arrives.  Especially if the freezer is working, so you don't have to deal with those items.

My nearby grocery sells dry ice and it has those cheap blown styrofoam coolers on top of the frozen food chillers.  I know all this from the last time I had to do a refrigerator repair.