I am not that familiar with the monitors that Nintendo used.
If you are working on the sound amp, there is no need to discharge the monitor. They are separate.
Why do you want to replace the monitor? At this point, you're saying the sound amp has a problem, the cab itself is not worth salvaging, and the monitor needs to be replaced.
At this point, you just get a different cab and swap the game PCBs into it since nearly everything worth having on the original isn't worth saving, from the info you are giving.