Once the base is off, you will see a circuit board held in place by 2 very small screws, shown here circled in yellow. It appears there are actually 6, but the other 4 larger ones are holding the optical sensors underneath the board. You will also see the 4 metal ball weights. Remove these immediately so they don't fall out accidentally. Note: If you don't like these rattling around, use some paper to tighten the fit when you assemble the joystick.
Be VERY careful with these 2 small screws, they strip easily. Remove the screws and carefully lift the board up.
There
are 4 connectors attaching wires to the board, three of which can be unplugged
(circled). Un-attach the wires coming from the joystick handle and the 2 optical
pickups (the flat-nose pliers work well for this).
Put the board aside (The gray-green joystick cable will still be attached), keeping the clear optical sensor on the underside protected from dust.
Note on Mainboard buttons: If you have a malfunctioning button, you can replace it with one of the 4 base buttons on the mainboard (I've never known anyone who actually uses these...). You'll have to be good with a soldering iron, of course. If you need more details, contact me. Note that while you can rig a base button to replace a hat button, it won't work well, or last long.
UPDATE: If you want to keep your base buttons, you can probably get the buttons from an electronics supply store. They are made by Omron. As far as the hat buttons, they are Omron part# B3f-3122.