sounds like the ever-so-common problem with the power supplies these shipped with.
with the machine off, slide it around so you can get at the back door. open the back door, and to the left you'll find a wood mounting board, and attached to it will be the larger game board (that the JAMMA harness plugs into) and a smaller sound board (connected by a ribbon cable and a 6 or so wire harness) and below those will be (most likely) a grey power supply that says PETER CHOU on it.
the reason I say this is because a ton of Midway games during this era came with that power supply, and when they start to fail they do exactly what you speak of where everything else that's powered by 120V AC (the marquee, monitor, in the case of some games even a cooling fan) will turn on, but the game won't. the power supply is stuck in an overcurrent protection mode and just remains in shut down.
sometimes you can get lucky and with a few power cycles every 15-30 seconds, the game will power on, and continue to work fine. others, well, inevitably in either case you'll have to replace the power supply.
fortunately it's not a difficult task, just might come off as a little tedious and one of those things that you're more-so afraid you'll mess it up than anything else.
screw terminal type power supplies are easy to figure out. they'll have a handful of different voltage rails, for MK2 the required voltages are +5, +12, and -5, then your black ground wires will attach to the GND lugs, and you'll also have the AC wiring that powers the power supply unit itself, which if they're marked hot or neutral, make note of that to carry over to your new power supply, and then there's the green earth ground wire, which is marked by an arrow-pointing-downward symbol with 3 lines going through it. (the down arrow signifies that you're grounding it to earth, make sense?)
where do you get a replacement power supply? users here prefer
www.therealbobroberts.net, I personally (as a tech in an arcade) get mine from Happ, the Happ PowerPro is pretty awesome. if you go either route, make sure it's a screw terminal unit, you don't want anything molex unless you want to splice wires.
once you reach this stage, PM me, and make sure you have a digital multimeter handy. I'll show you how to adjust +5 on these.