The most straightforwards way to do this is send a message to the phone using HAYES AT commands. You would need a programmable bluetooth module to be triggered by your button.
Air pressure is the result of gravity pulling down on the atmosphere and stopping the air from drifting off into space. So at sea level the air pressure is about 14.7 pounds per square inch, that's the weight of a 1 square inch wide column of air from sea level to the top of the atmosphere...