By all means try the simple stuff first - reseat/replace RAM, reseat CPU card, airport card (+ harness), and make sure the Cube has a good PRAM battery.

If this fails, the random switching off could be due to a faulty reed switch (the touch sensitive power button); this is a common issue and I recall Cubeowner forums suggesting some simple fixes. If this fails, it could be the VRM, so you might want to check the lines with a multimeter in this instance.

JB