I have transported dozens of machines on their backs and sides in my van, no problems.
If the machine is in such a state that the monitor mounts are that worn or rusty, you shouldn't buy it anyway.
You can't "prop" a monitor up (set the monitor inside the cabinet without securing it) on most arcade machines, so it should be bolted to the frame. If you are worried about it, take a ratchet and screwdriver set to the place you are buying it to inspect and secure the cabinet. Make sure everything is reasonably secure (boardset, marquee, monitor glass and bezel) before you transport. This process should take you about five minutes.
In my experience, It is incredibly rare to see anything come loose during transport that wasn't obviously loose to begin with.
APf