Don't worry about the CRT, you only have to discharge it if you are removing the anode cable (big red wire to the "suction cup"). Take care not to bang around the neck board plugged into the end of the CRT. The evacuation pip is delicate and it's all to easy to hear that sickening CRACK-PSSSSSSSSSSSSsssssssssssssssssssssssssssssst and the tube is done for.

Caps are definitely recommended, but if it works fine, you can probably leave that alone for now. I do advise touching up the solder joints holding the deflection yoke connector to the analog board though, they are a common failure and loss of deflection can burn the phosphors in the tube.