first thing I'd suggest doing is pulling out the frig to check if the coil behind the cardboard at the bottom is caked with dirt, and see if the little fan is running down there. If the coil is dirty clean it, the compressor is thermally protected and a plugged up coil or shot fan will cause the compressor to overheat and shut off until it cools down.

Other than a dirty coil or shot condenser fan any other problem would be inside the refrigerator. The problem could be a bad defrost control board (common), a bad evaporator fan behind the plastic in the freezer (common), a bad defrost coil (rare), or a bad thermal sensor (rare) attached to the evaporator coil. All except the freezer fan are part of the defrost cycle, and it's easy to see if you have a defrost problem by popping off the plastic cover in the back of the freezer compartment to see if it's completely compacted with ice.

If the compressor coil is dirty it's an easy fix you can do. If the compressor fan, freezer fan or circuit board is bad it would probably be worth having it fixed because labor costs would be minimal . A bad defrost coil isn't a quick fix so labor costs may be higher than you're willing to spend on the old frig.