Why not use a transformer inside the monitor? You could tap the 120 VAC leads, shouldn't really strain anything. Using a small ITX powersupply or laptop powersuppy would give you a few watts to play around with. This would be good for things like 12V fans, LEDs, Cathodes, etc. As for working on your monitor, it is very dangerious, but I'm assuming you're going to take precautions. The first is to just let the tubes discharge by themselves. Generally this means letting your monitor sit unplugged for a week or so. You can also buy a discharging unit or try making your own, and grounding the tap at the top of the tube. Depending upon the model you might also have to ground the flyback.
Also, be careful that when you add fans or electronics that it doesn't cause harmful interferance with the signal, nor shorts anything inside.