The first thing I would do is to label all of the wires connected buttons and joysticks microswitch. Use white electrical tape and a sharpie to mark which buttons are which for eg PL1B1 for player one, button one etc.
There should be a common ground daisy chained around the panel so I assume you would be a able to connect any of the ground disconnects up to any common terminals but to make sure and things super easy and neat when you reinstall everything I would tape up the corresponding ground disconnect with the NO switch disconnect.
Once you have done that you can remove all of the disconnects from the button/joy micro's.
To remove the buttons you have to remove the microswitch buy carefully pulling one of the prongs out of the micro and then the micro should be able to be removed. Once you have seen/done one its a doddle. Once your micros are removed you just have to undo the nut and the button will slide out of the top of the panel.
For the joysticks its hard to say but it doesn't look like they are held in by carriage bolts from looking at a couple of random slickstik images I could find online. I guess they use flush mounted t-nuts in the top of the panel and machine screws to hold them in place from underneath. It should be fairly easy to see how you can remove them.
If you are not sure, post back with some pictures if possible.