Not knowing your talents for appliance repairing, I will give you the items that require very little expertise.
1. Check that the door seal is closing all
around both compartments. The piece of
paper test works just fine.
2. Your evaporator is iced. make sure your
drain trough is clear of any obstructions.
3. Do not put food near the thermostat that
will block air flow.
4. Leak in the refrigeration system. Check
all of the copper lines for small leaks.
Any other check will require the use of a VOM plus some knowledge of the component parts or the ability to read the schematic. If you want to go that route let me know,
otherwise you will have to call the landlord.