Well, it's possible, but I wouldn't recommend it.
Finding a replacement monitor for this should not be too hard. Do you have arcade or amusement machine supplier in your area? You can call them up and see if they have any 19" monitors around. Or you could look on craigslist for other cabs with 19" monitors and see how much they would want for that. Or you might even find a user here that has one they'd sell to you.
If you don't want to use an arcade monitor then I recommend using a pc monitor. It's closer to an arcade monitor in image quality and easier to work with.
BTW, all caps = I'M SHOUTING EVERYTHING I SAY!!!
Truthfully, since this is your first post on here I'd do this if I were you:
#1. Take out everything inside the cabinet except the controls, speakers, lights, and coin mechs. Put it up into storage somewhere clean, cool, and dry. After you get into this hobby a bit more you'll want to revisit it, but leave it for now.
#2 Purchase a 19" pc monitor and a budget computer. You don't need anything fancy. Anything over 300 mhz and 256 mb of RAM should work.
#3 Install Windows XP and MAME on your computer. Set MAME to run Pac-man on startup of the computer. Also set windows to automaticaly log in.
#4 Purchase an i-pac from ultimarc.com. Configure it within MAME.
#5 Install the monitor, pc, and i-pac inside your cab. You will need to build a little bracket inside the cab for the monitor to sit on.
You're done. You have a pac-man cabinet.