This seems to be somewhat similar to your problem.
If your N900 turns off randomly, then it's hard to pinpoint exact place where hardware needs to be repaired.
If it turns off when the screen turns off, then it should be easier to solve. Maybe, there is a short-circuit of some kind related to display and battery charger?
I know of a computer with opposite problem: once the computer is turned off, its display (graphic card) becomes disconnected from the computer, and the computer needs to be warmed up to get it to turn on with working display. The hypothesis is that when it warms up, the metal parts become connected again.
Best wishes.