As I understand it, there are two types of pressure tanks; one has a bladder and the other type has an air makeup valve and is air over water. Each of them must have a way to get air into the system, but each differs in how it is done. On the bladder tank, there is a valve similar to your tire. You can check and adjust the air pressure as you would a tire. You must check this with the pump off and a valve opened to allow the pressure tank to reach equilibrium. then you can adjust the pressure, if required. This type must be filled to 1-2 pounds less than the turn-on pressure of the pump pressure switch.
The other type, since the air is exposed to the surface of the water, will disolve all of the air. You need to turn off the pump, drain the tank so there is air in it, then you can turn things back on. As your pump fills the tank that air is trapped and becomes pressurized by the pump, and will push the water out, maintaining the pressure until it drops to where the pump turns on again. If there is no air in eithertank, then since water doesn't compress, it will quickly reach the turn-off pressure, but since there is no air cushion to provide the pressure storage (like a spring that is compressed), as soon as you open a valve, the pressure drops radically and the pump turns back on.
If your tank doesn't have any kind of air valve on it, it doesn't sound right to me. The air-over-water types usually have an air make-up valve that will let air in, but prevent air or water from getting out. The bladder tanks, when the bladder fails, will fill up with water. If the valve is defective, over time, the pressure can drop, too. Note, on a Schrader valve, the primary seal is the cap, not the little air valve itself. My unprofessional experience.