I can't tell you EXACTLY what to look for insofar as that particular display is concerned, but I can provide a general idea.
First, my disclaimer: I assume that you know and understand the safety precautions required for working near a CRT, and the consequences of not following said safety precautions. Otherwise, use Google and look up such terms as 'CRT discharging' and 'capacitor discharging'. I wish you luck and hope you don't get hurt, but if you do, I can't take responsibility. I've done this sort of thing many, many times and have yet to be shocked. Just don't touch anything that you don't absolutely have to, and you should be fine either way.
Now then, assuming that your problem is similar to those experienced with other Apple products (17" Apple Studio Display (aka Mitsubishi DiamondTron), original iMac G3), your problem is most likely either a bad component or dry solder joint in the HV circuit.
The first thing to look for (after accomplishing the mandatory safety procedures) would be loose connections. Anything majorly loose would be readily apparent, but look around for anything awry.
Continuing on, you would then be looking for explodey components, most notably capacitors or variable devices. Since physical shock restores functionality to your display, I doubt this is the problem, but it never hurts to check.
Finally, check out the solder side of the circuit boards on there. Examine especially closely near the flyback transformer (danger: no touchy without discharging first! Those things contain up to 20,000V) and other large components and connectors such as capacitors and transformers. Dry solder joints may be difficult to see, so you may wish to employ a magnifying glass and a fair amount of patience in your inspection.
Hopefully one of the above steps finds your problem. Dry solder joints have created havoc in many, many things for me (from my Civic to my Apple Studio Display to my Technics receiver), so I'm no stranger to a rework.
Good luck!