Of course, you'll need a couple of solenoid valves (or a 3-way valve) to control air-flow in & out, and you'll need a relay or some other electronics to turn the air pump on & off. (You can't directly power valves or a motor/pump from a microcontroller.)
The logic simple... The logic is just like a thermostat... You turn-on the pump until you reach the desired pressure, then you turn it off. If you want to reduce pressure you open the release valve until the pressure drops to the target level.
An MPX5010 (or similar) would be a simpler choice than the SPD005G suggested by jremington. What you're going to find is that some pressure sensors (like the SPD005G) do not provide an amplified output; you get an output voltage in the ~0mv to 20mv range. Devices like the MPX5010 provide output voltage in the 0 to 5V range so it can be interfaced with an Arduino with no instrumentation amplifier required.
Regarding PID controllers, consider them simply as a clever output control. Right now you're probably imagining writing code that has a basic "if the pressure is less than X turn the pump on", and that works fine for the most part. In practice you'll probably find that the pump being erratically turned on and off with code like that; physically there's just not a perfect correlation between turning the pump on and getting additional pressure. A PID controller learns this difference between the motor being on and pressure increasing and manages the pump in a more stable manner.
Pneumatic solenoids shouldn't be required with a pump like the TYAP1212. If you were going to use the pump to pressurize and air tank to use as a supply you would need them but I doubt you're going to do that here.
couple of things.
PID is for a variable control.
pump on and off is not variable. it is on/off
if you have a pump that fills a tank, you can use something like a needle valve that can allow variable flow into your bladder.
that would mean a pressure sensor for your holding tank, that would control the pump to come of at some point lower than some setting, and then turn off once it got high enough.
the needle valve could adjust the pressure in the bladder.
if you want to control for an athlete that is running, then you would need to do something much different. the expansion and contraction of muscles would effect the bag volume and according to Charles Principle, altering the area would effect the pressure.,
as for the multiple pressure settings, all that is very easy once you have worked out the flows and pressures and volumes needed for your application.
google sphygmomanometer pump to get an idea what might work.