macbook 1,1 shut down prompt pops up
The MS guy in you would be right - best thing to do to hand a Mac to a new owner is nuke &amp; pave - though strictly it's step 5 - What to do before selling or giving away your Mac. 

The latest OS that machine can run is 10.6.8 which is still available from the Apple Store [it's back from the days when you had to pay for the OS] so there's no real way round that; it won't boot from USB either, it's too old.

AHT should be on the disk too, &amp; that's where you'd need to run it from. It's only built into the machines from much later.

Nuke &amp; pave in itself may not fix the issue, but it would at least eliminate all but hardware as the cause.