|Do you ever see "anything" on the screen?|
If it shuts down before even attempting to load Windows then it must be a hardware fault. Could be any number of things, so it's a matter of how you feel about poking around inside - not so easy on laptops.
One quick thing you could try (not too hopeful) is to see if the RAM sticks are accessible under a flap somewhere. If so take them out and clean their edge connectors with a pencil eraser and pop them in and out a few times to clear any oxide off their sockets.
Depending how far you want to go it would be worth ensuring all the connectors you can see are firmly in their sockets.
Keep watching - someone might know that particular computer better than I do and be able to suggest something more.
Always pop back and let us know the outcome - thanks