There should be an e-clip on the base of the joystick that holds the actuater (plasitc square piece that activates the microswitches on the joystick) and a spring in place. Pop that off and you'll be able to pull the joystick right out. Just reverse the process to put them back together. The only thing to be aware of is there is metal washer looking piece that's inside of the plastic base which the joystick needs to go through. With the joystick removed it will be able to slide around a bit in the base, so when you go to put the joystick back together, you may have to shake the base around to get that metal washer back in line again.
By the way, I'm working on restoring a Simpsons cab myself. Are you going as far as replacing the artwork or is yours still in good shape?