Well, I have thought of doing this as well.
There shouldn't be a problem with the monitor being out of it's housing as long as it is insulated. There is the danger of shock though, and exposure to the elements of your house or garage.
I think it is possible to re-use an old television case for this purpose. One of the big wood grain RCA tv's would be a great choice.
Something along the lines of this might work for a casing.


An RCA Lyceum TV
This particular model would work well for 25" monitors. 27" cases are common as well.
If you go to your local television repair place it is likely they will just give you your pick of several dead tv's or cases. My local place had 3 sitting outside the front door that he said I could have if I would just get them out of the way.
OR you could simply build a box for it. Shouldn't be too hard if you re-use the chassis from the arcade cabinet.
OR
you could buy one already made. I haven't dealt with MAS systems in a long time, but they were a very reputable dealer with a loyal following for many years in the fighting game community. They sell arcade monitors that have been consolized.
http://www.massystems.com/RGBMonitor.html