its possible. They make devices that are used to turn lamps and stuff on and off automatically. They are like small little clocks with a plug built in. http://www.rewci.com/inaptimfresh.html
I am sure you can use this with a computer as well. If you short out the on/off button connectors on the mobo (with a spare jumper or some foil why not) and you set the timer right, when the computer is plugged in, it will automatically turn on. The timer manages the power to the computer so it will "plug it in" at the time you set.
That being said, I am not sure how the computer and Bill's little brain child will react to the on/off button being held down non stop. I am not sure if it will boot, and shut down infinately, but i think that it waits for the switch to be released before taking another input from it.
If my theory is correct, if you short the on-off button connectors on the motherboard with eachother, (imitating a push and hold of the button), and you set the timer and plug the computer into the timer, when the time hits, power will flow into the computer and you can have a program shut the computer down when its done.
note, programs that restart the computer may affect the entire operation, so those wuold not work.