The answer is yes, and no.
Yes it can be done, but you will spend more money doing it than an arcade monitor would cost.
You would need a scan convertor that supported rgb, ground + sync. That would cost more than an arcade monitor.
Only exception is some of the OLD Amiga and PC monitors (not the kind with VGA plugs, the kind with the plugs that look like serial port plugs.), which will work just fine, only thing required is some dodgy wiring. But only SOME of those work. Offhand I think I remember that the Amiga 1080 monitor works.
Anyway, you DON'T need an extra monitor to troubleshoot what is obviously a monitor problem. All that is going to do is tell you that there is something wrong with your arcade monitor.
Guess, what, there is something wrong with your arcade monitor.
There are many troubleshooting guides for arcade monitors on the net. Tubes are almost never bad, so I suggest doing a cap kit, and replacing the flyback. Chances are good that those two will bring it back to life just fine, and it will probably never need another repair.