this is what Alternative Modding is for, to answer questions like this, heh.
Anyway, the only way to spray-paint the whole monitor would be to take it apart. On the back of the monitor are ventilation holes, and while you can cover up the screen, you can't cover up those holes without taking the monitor apart.
You will definately want to take your monitor apart before you paint it. But, be VERY VERY careful because the insides of a monitor can kill you. what you will want to do is probably take it to like a TV-repair center and ask them to power it down. The capacitors in a monitor can store energy for quite a while, and they store a ton of energy (enough to kill you). If you don't want to take it somewhere, you will want to unplug it, turn it on and off a few times while it's unplugged, and then let it sit for a week or so. that SHOULD be enough time for all the power inside it to dissipate. I'm not making any guarantees though, heh.
Bottom line, you're not going to get a good coat of paint without taking it apart. And, taking a monitor apart is very dangerous. If you do it yourself, be careful. unplug it and let it sit for a while. You can just be ultra-careful and just don't touch anything, but you will need to touch the parts at some point, heh....