Do you mean it just goes off - POW - the very same moment you press the up arrow, or that it goes into a normal shutdown?
Is it a laptop or desktop? If a laptop, I would suspect a loose connection or faulty component right under that part of the keyboard. If a desktop, is it a USB or PS/2 keyboard?
You posted this on CNET Forums too.
On there, Supermaster wrote,
Your keyboard has probably become defective. Check with some other keyboard whether the problem persists or not.
That is good advice. Try another keyboard.
Although this is very strange. I do not see how the same keypress could shut down the PC in both Windows and Linux.