I'm assuming that you have already been through all the jazz of making sure that your monitor is not turning off because of some "power saving" setting. The last CRT monitor I had would turn off after a certain period of non-use. However, I'm guessing that you have already established that this is not the problem.
By no stretch of the imagination am I saying that "CRTs cannot be fixed, period", but it has been my experience that buying a new monitor (perhaps a flat panel) is usually preferable to trying to fix power problems with a monitor, if the problem is truly with the monitor itself.
If I were you, I'd start by plugging the monitor in to another wall socket, or power strip. Frequently, "bad power outlet" is the problem rather than "bad monitor".
In the event that you narrow the problem down to being "the monitor", I generally advise simply getting a new one. If you don't have any experience working on the inside of CRTs, you're probably going to need to get with someone who does -- numerous special tools and safety procedures are required, and parts may be difficult to find on your own. If you choose to have the monitor fixed by someone in your area for a price, be prepared for that price to be as much as a reasonably good flat panel.
Again, not saying a CRT monitor can't be fixed, its just that more often than not, it isn't profitable to do so.