How do I schedule waking up from hibernation?

When hibernating, the operating system is off and can't do anything. It's the computer's BIOS that does the waking up, so it's only possible if your computer's BIOS supports it. With some BIOSes, you can configure a scheduled wakeup quite easily.

Press the key that gets you into "setup" or similar when the computer first boots up (often F2 or Del) and see what options you have.

Alternatively you may be able to configure the BIOS to wake from Ubuntu, if the BIOS supports waking and allows the OS access to configure it. This guide is quite comprehensive:
http://www.mythtv.org/wiki/ACPI_Wakeup

I found that it is difficult to wakeup Ubuntu from hibernation. So I use the following commands to boot Ubuntu at a predefined time:

# Clear previously set wakeup time 
sudo sh -c "echo 0 > /sys/class/rtc/rtc0/wakealarm"

# Set the wakeup time at 2:02 am
sudo sh -c "echo `date '+%s' -d '2am next day + 2 minutes'` > /sys/class/rtc/rtc0/wakealarm"

#Check if alarm is set. It should show the unix time stamp
cat /sys/class/rtc/rtc0/wakealarm

#After setting the time, PC can be turned off with this command
sudo shutdown -P now 

This setup has been very consistent and it has never failed. Auto-login should be enabled during Ubuntu installation for this to work, or else Ubuntu will wait at the login screen when it auto boots up. More info here.