You have a couple of choices. The easiest is to buy an ArcadeVGA card and hack a VGA cable to connect to the arcade monitor. Look for details on how to wire it on
http://www.ultimarc.com/monfaq.html - They also sell the ArcadeVGA card. You'll need to spend some time with a multimeter and possibly a soldering iron to do it, but it is rather easy. With the ArcadeVGA card, you can use Windows MAME or DOS MAME, or just about any version but AdvanceMAME.
Another choice is to use nearly any VGA card with Advance MAME. The Trident Blade T64 is often preferred, and is available new for less that $40. I like the SiS 6326 card because I can find them used cheap. There are many other VGA cards that will work. See
http://easymamecab.mameworld.net/ for a great way to get started with Advance MAME. If you have (or can get) the compatible hardware, their "Easy Cab" CD is the easiest way to get Advance MAME up and running.
I have made a couple of cocktail cabinets. Mounting the arcade monitor was the least documented problem I had to solve. On my Midway-style cabinets, I used pine 2x4's to hold the monitor away from the side of the cabinet, and hardwood 1x2's to hold the monitor the correct distance from the cabinet top. Thick washers were inserted to adjust the spacing, and large lag bolts were used to hold everything securely.
BTW, you can also get a refurbished Wells Gardner 19" monitor for $145.