Hi,
Wow, unless you have experience working on high voltage supplies i dont recommend you trying to fix this. The anode has very high voltage on it and im not talking 200 or 300 volts, but 20000 volts or even higher. It's very dangerous not something to fool around with. You have to know how to discharge it before working on it or you might not be around the next day to fix it anyway.
That said, this is really a tricky problem where you probably need a schematic unless you can spot something burnt on the circuit board.
There's a slow start circuit in there somewhere but if the cap went bad...oh wait a minute...if the cap went bad it might be trying to start up during the first 10 minutes but not successfully because it may be trying to start too fast and the over current sense circuit turns the high voltage off thinking there is an over current. Eventually it builds up enough charge to allow it to start more quickly so the over current sense doesnt trip. Still, you'd have to locate this cap *if* that's what it really was to begin with. That's a bg *if* too as i dont have the thing right in front of me.
Did you have this thing apart yet? (not that i recommend it though unless you have a lot of experience with this stuff).
I dumped all my CRT's because the LCD's that replace them take up MUCH less room and use less power. Prices have come down a lot too in recent years...maybe time to think about a new LCD (LED backlight of course).