I did that with A Little Rage, and it turned out pretty good.
Heres how I did it.
Mounted monitor and plexi. used frog tape to tape a rectangle that marked the screen part of the monitor. Used frog tape because it does not leave a residue, paint wont soak through, and its easy to remove.
Removed plexi. Flip it over and frog tape the inside of the rectangle made by the frog tape on the other side of the plexi. Filled inside the whole rectangle.
Taped outer edges of plexi to prevent paint from spilling through to the other side.
Spray painted the plexi with black. Alowed to dry, put on another spray coat. let it dry overnight.
removed frog tape, flipped over removed rectangle frog tape.
Inserted plexi back over monitor and it looked great.
The only thing I don't like about this method is that if you want some art on the bezel around the monitor, its near impossible.
Maybe place the cut out art on the plexi with some kind of clear glue, then paint over it? Or stick on art on outside of plexi?
Link below in my sig if you want to check out finished product.