The PSP doesn't have video out as an option.
You would need to use one of the PSP TV adapters out there, and then interface the RCA outputs (which all the TV adapters I know of use) to an arcade monitor. You could possibly use a VGA box to an arcade VGA... maybe? But you'll be going through quite a few gadgets and the end result will probably look pretty bad.
And even then... there are no widescreen arcade monitors, so you'll lose a good amount of screen real estate. And the resolution would probably look ok on an arcade monitor considering its already low resolution, but the PSP really isn't meant to be seen on a 19" or 25" screen. AND then you'd have to interface the controls somehow.. a lot (at least a few off the top of my head) REQUIRE the use of the analog nub, and NOT the dpad. That analog nub will be difficult to impossible to implement into a control panel.
I guess my real question is: why on earth would you want your PSP on an arcade monitor?