I believe that green button controls pneumatic valve(s) for the main piston, as well as two small pneumatic clamps.
It could be an electrical button in which case valves controlled by it are most probably some solenoid controlled valves (i.e. they open or close with electrical signal).
It could also be just a push button of the pneumatic valve that is controlled mechanically by that button, and the rest of it is inside the control box.
I believe this is more likely than previous case.
Starting and stopping of the drilling motor is probably controlled by some electro-mechanical limit switches that get mechanically triggered by the head traveling up and down somewhere along the way.
Your pneumatic cylinder is double acting so the problem is with the air supply at the bottom side of it (that's why it's not going up). I would suggest opening that control box and see what's inside, or try to peak from underneath it.
Just follow the air hose from the bottom of the cylinder backwards to the control box and see what gives it its air.
The cheapest try would be to replace that thing first and see if it helps (I assume you don't hear any air leaks at this time and some hose is not pinched somewhere so that air can not flow through).
If possible try to read the name of the part, find out what it is and than find locally available equivalent replacement. Save yourself money and time by not trying to find the original part.
Local rep from www.smcusa.com was extremely helpful in my case.