I have left newer games running for 4 months non stop with no problems.
The old stuff is very reliable IF it is in good shape, but most of those caps are now what, 25 years old? If the electronics have been recently serviced and the game is mint, you can let it run like it did when it was new.
Most of my failures on these old machines have been on start up, most electronic failures are on startup. Ever see a lightbulb blow when it's on? It goes out most of the time when you turn it on, not while it's running. Same with electronic devices, especially caps and transistors.
Old games may not have the screen saver ability of new ones. So you can burn in the screen for no reason if it's your personal machine. That and power consumption would be the two reasons to shut it off.
I shut off all of my old games in my personal gameroom.
If it's hard to shut off, then go to walmart and check out a remote control for it. They have an outdoor christmas light remote this time of year. They are like $12 bucks. They are used to shut off outdoor lights without having to go outside so the range is really good. They work great indoors, and they support 12.5 amps and 1500 watts so you can hookup more than one thing on it.