Before fucking about, please discharge the tube. There's a suction cup with two metal contacts touching the tube.
Inside the TV there is this one chip or similar. Look up on The Internets to find out. The chip has a marking.
Read it's spec sheet.
Video out is the output signal of the RF de-modulator, that's your other source. The wire connecting to it is goes to the CRT driver. LITERALLY just wire the composite to that one wire/trace and you should have a signal.
For audio, wire there should be a trace corresponding to the audio output. Do the same as you did for the video out, cut it and wire it to the input of your composite audio.
If you didn't mess up anything, you should be able to use your monochrome CRT for gaming.