I'd say the best thing to avoid painting the screen itself, or getting paint on the inside would be to completely disassemble it, separating all components from the plastic casing. Of course, this does expose you to high voltage risk from both the circuit board and the CRT itself. There's also the danger of damaging the heavy CRT (where most of the weight of the monitor lies), which, if it implodes, can send lead-laden glass flying everywhere at high speeds. The CRT's neck is especially fragile. But, doing all this will ensure that no paint ever hits the screen, or worse, internal electronics.
I've not ever had a monitor apart for painting, just for repairs, and it's not easy, especially moving around a ~65 pound hunk of easily scratched, somewhat delicate, vacuum-sealed glass. And I don't know if that'd be needed either - but I'd think that masking tape on the picture tube would be 1) a bad idea, and 2) inadequate.
But that's my take, from someone who's never painted a monitor. Though actually, I probably could have, since I had the thing in pieces to find a
bad solder joint. Would have had a nice green monitor to match my PC.