My Sony Trinitron picture flashes and the sound goes out when the screen flashes. It's about 20 years old and I want to know if it's worth repairing. It has a great picture and sound, when the flash isn't happening, which is very rare.

I would guess a bad solder connection(a hairline crack around conductor to solder pad),it is common to audio/video, narrows it down.
A good Visual inspection of board may reveal.
Old Techs are more rare now on tube sets.
New TV would make a nice Christmas present to yourself.