replace c31 right off of the bat and go from there. Its a often bad cap. Use a higher voltage but same value to make it last longer. I have seen bad ram take down the power quite frequently also. Most logic could be suspect and can be tested in circuit. Google the ic specs to find ground and test from there. Easy peasy.
Can try the 'hot chip' test....leave power on for a while and check with a temp probe or a flir camera.
Its usually c31 though. they only last 10 years at best. Newer ones are made better.