Have a look at your Computer's BIOS. Most BIOS have the feature (RTC) to turn ON the system at a given time. I have been using this feature for quite a few years now.
My PC does have this ALARM function. I setup the alarm, but the PC did not boot at the set time.
Should I be shutting down the PC in any specific manner. I did a standard `shutdown -h now`
Thx Sans