The wall fridges are unique and cool to have in any kitchen! They have quite a few unusual aspects to them. Because they are more or less a built-in appliance, people will go to great lengths to get them repaired since replacement isn't an option and the kitchen isn't complete without them. Due to the fact that there are few people who proficiently still work on them; many have had horribly botched repair attempts done out of desperation.
Because of that, you have to be extremely careful to do thorough troubleshooting and leave no stone unturned. As John said, the early ones often had wear-out problems with the compressor. Many have newer compressors installed from "back in the day." Many of the replacements have, it seems, begun to lose performance as the decades passed.
The most recent one I worked on had a fully blocked capillary tube due to a bad filter-drier. It caused the compressor to run but not cool; and then to act as if it were seized up. In reality the compressor was still perfect, but the blockage "outwardly" emulated a failure of the compressor, when a technician did not take the step of using his gauges to make a proper diagnosis.
The design of these fridges is such that the refrigerant first cools most of the freezer, then cools the fridge compartment, and then returns to the freezer where the suction accumulator is located. If the refrigerant charge is reduced, it will first stop cooling the refrigerator cabinet evaporator coils, before the freezer loses cooling.
The refrigerator part contains the one and only thermostat. Once the charge is low, the fridge will run all the time, resulting in an abnormally low freezer temperature and poor fridge cooling.
There are copper-to-aluminum joints inside the cabinet where the tubing connects, as well as braze joints on the condensing unit. It would be prudent, in my opinion, to start with visual checks and be sure there are no obvious oil leaks on any lines and to be sure the fan is running and the coils are clean. Then from there, take the upper right cover off, inside the refrigerator cabinet and inspect or "sniff" the copper-to-aluminum joints for refrigerant leaks.
The bottom line is, leaks are fixable, but some take a lot of time and effort to find and fix. It is very likely you will need to have a specialist make the repairs unless it's an obvious leak that any qualified technician can repair.
Hope this is helpful, and good luck getting it going!
Sincerely,
David