first try to clean the condenser coils ( the lower black radiator down by the compressor)
unplug the machine, & brush off all of the dust-bunnies from the front & blow it out with compressed air if available, try not to over-spin the fan motor ( it's not a turbo charger but it will sound like one if you over speed it with the compressed air & you may damage its motor, temporarily stuff a rag in it to keep it from spinning, just don't forget to remove it before you plug the machine back in!)
Be very careful if you decide to try to power-wash it or use the garden hose on the condenser fins, the old fan motors really don't like to get wet, it may trip your GFI or your circuit breaker if it does get wet, so put an over size baggy & tape it up before you hit it with any water, again with the machine unplugged of course!
Oh & If the fan you are referring to is also next to the compressor, it's called the condenser fan & yes it will cycle on & off with the compressor, the evaporator fan is up top inside with the soda's & runs 24/7/365.