There are 3 different types of possible breaker trips. The solution will vary depending on the type of breaker trip.
This is simply "too much stuff running at the same time on one circuit". This should rarely happen in a properly wired kitchen. However, many older kitchens were wired under older versions of the NEC, and before modern appliances such as microwave ovens. A typical microwave oven may use 1200 watts of power when running. If it is on a 20A circuit (the standard these days) then that is only about 1/2 the circuit capacity. If it is on a 15A circuit then that is about 2/3 of the circuit capacity. A refrigerator may use 500 watts or more, so the total can easily exceed the capacity of a 15A circuit, particularly if there are other devices on the same circuit.
The solution to this is to put the microwave oven or the refrigerator on a dedicated circuit. There is a different rationale for each one: Putting the microwave oven on a dedicated circuit makes sense because it uses 1/2 (possibly more) of a full circuit, leaving relatively little capacity for other things. A refrigerator typically uses far less than 1/2 of a circuit's capacity, but it is very important that it always be running, and a dedicated circuit means other things (like your microwave) won't stop the refrigerator.
The problem is that adding a circuit can be anywhere from cheap and easy (space in the breaker panel and a short run from the panel to the kitchen) to expensive and hard (no space in the breaker panel and/or a long distance from the panel to the kitchen).
- GFCI - Ground Fault Circuit Interrupter
This is a device which protects people from certain types of electrical problems that can kill them. It is a very important safety feature around water, so it is now required for most circuits in a kitchen. However, an old refrigerator can have certain types of problems that result in a GFCI tripping but which don't present a real danger to people. In addition, a legitimate GFCI trip (which may be happening with your microwave, for example) might be unnoticed, leading to a refrigerator stopping and spoiling food.
- AFCI - Arc Fault Circuit Interrupter
Arc faults are more of a fire danger than a direct life safety issue. Large appliances (both microwaves and refrigerators) could trip an AFCI under some circumstances where there isn't a true/dangerous actual arc fault.
The first step is to figure out what type of breaker trip you have. If the breaker has a TEST or RESET button or any indicator lights on it then it is most likely a GFCI, AFCI or both. If it does not have any buttons or lights then it is most likely only an overcurrent breaker. If it is a GFCI or AFCI breaker then the next time it trips you need to figure out which type of problem caused the trip. The method of figuring that out varies by breaker brand, but we can help if you upload a picture of the breaker.
Overcurrent can really only be fixed with new circuits, but the usual workaround is to move things so that there isn't too much stuff on any one circuit. If you can do that without extension cords, great. If you can't do that without using extension cords then your landlord should provide a fix - generally a new circuit.
GFCI and AFCI are indications of actual problems. They should not be ignored. If a microwave oven causes frequent GFCI or AFCI trips then that is an indication that it is faulty and should be replaced.