fwiw;
Most of our tpa bay area shallow well pumps around here turn Off around 40 to 45 lbs and turn on , I dunno- On at maybe 30 lbs. Some are higher but really not too many in small or typical residental sprinkler system applications.

Anyhow - that little pressure tank has a tire valve on it.
If its lost pressure all kinds of odd starts & stops happen.
Pressure is supposed to be just slightly under, by about two lbs of, the cut-in or On pressure.
If it will still hold air - You check the pressure after killing the power and letting the water pressure drop to zero or simply let the water run out of the tank side (which could cause it to lose prime if leaky check valve or air leak on the suction side).

Maybe you might call your landlord before doing too much and get his approval before you tinker, in case something is bum & you dont want to be blamed for it.