I often have to section the back doors of Mame cabinets for the monitors to poke out. The worst of which is my Amazing Maze, which has a monitor that pokes out about 3".
In the case of a one or two inch variation it is pretty simple. You just make the required cut in the back door, then surround the cut with a 2x4, and then put ply wood over it. If it is then wise to make an identical protrusion near the bottom so that the game can still be laid flat on it's back.
It isn't just Mame cabinets you see this done on. My first Turbo was modded that way when I got it, and Dave's Robocop is as well. (Replacement monitors don't always have the same size frame).