The cap is like any capacitor, only bigger. It would slowly lose its ability to hold a charge, so yes, the game could work to a certain extent, but the worse that cap gets, the worse the quality of power coming into your PCB and monitor. It could be bad enough to cause problems on the PCB. If your monitor is blowing its fuse, definitely replace it. That is step #1, making sure you are getting clean and proper power on all lines (IIRC, and I probably don't, it's +12v, +5v, and maybe -5v, then whatever the monitor uses). Part of making sure that is the case is replacing that power filter capacitor.
Next you would test the PCB for proper voltage at its test leads. Those are the raised metal circles that are labeled with an output. There are definitely X, Y, and Z leads in the video section of the board, probably +5v and possibly +12v leads too. If those are all good, you know your board is working pretty well and you would test your edge contacts the same way. Once that is good, you would test the same things on the end of the video signal harness. Then you know you have good video signal. Check the voltages on the proper pins of the monitor end of the monitor power cable, and once you get that good, your monitor shouldn't blow a fuse. If it does, you have a problem on the monitor chassis, and you'll do a cap kit and start troubleshooting the chassis if that does not work.
Obviously, if you see anything amiss at any of these steps, you would correct it before moving on to the next one. If you're not sure how to do any of these things, we can work through them one at a time here.