"Doesn't come on" can be caused by all sorts of things. It could be something as simple as the interlock switch. Most games have interlock switches that prevent the game from powering on if the back door is off. If the door is missing, warped or damaged, it might not press the switch down. Similarly, if the bracket holding the switch is bent, it might not be lining up right. If you pull out the switch button so it clicks out - it'll make it stay on.
Check for a marquee light, humming or noise of any kind, coin door lights, any sign that the game is getting power at all. If it is, then investigate further.
The monitor may be working, but the game's power supply is bad, so it's not coming up - listen for whine from the monitor, look for neck glow, etc. Watch the screen while you switch the game off after it's been on for a few minutes, you should see a flash as the raster collapses.
But, if you want it for a multicate project, it doesn't matter if the game board works or not, or if the power supply is good even (you can use a computer power supply easily if the original is dead, and new arcade power supplies are only $20). The only thing you really care about is the monitor. If you don't want to learn monitor repair, then you'll want to see the monitor come on, or do something...
And, if it's a dead game, and it's cheap... can't be too picky - if the cabinet is good, then fixing the electrical stuff will be a good learning experience.
-Ian