Well then, it probably has a normal 8-way in it. There are snaps underneath that you undo by reaching in through the coin door. Once unsnapped, the control panel should lift up or swing down on a hinge (depending on the cabinet).
Once you can access the bottom of the control panel, you'll see an c-clip on the bottom of the joystick shaft. Pull that off, and the joystick shaft will now pull out of the top of the control panel. Then you can remove the four bolts holding the joystick base to the bottom of the control panel, and remove it. There may be a dust washer between the base and the control panel also, although sometimes you'll see it installed on top and it comes off with the shaft.
Installation of the new joystick is the reverse of the above. If you buy a brand-new joystick, it usually comes with installation instructions.
Obviously you will have to remove the wiring to the switches and move them to your new joystick. The best way is to take pictures of what wires go where, and maybe label each group as to what is UP, DOWN, LEFT, and RIGHT.