Using arduino to turn on computer

Hi,
I have a laptop that, due to poor available power, keeps on having problems with its power. What I'm trying to do is if there is no available power and the laptop's battery is running low to shutdown itself for 24 hours. If there is still no power, repeat this until there is.
Now, I can code python and getting the laptop's battery (etc) is trivial but I'm rather stumped at the part where the laptop has to suspend or turn on itself after 24 hours.
I was thinking of using Arduino because it is a robust microcontroller but I'm not sure (a) if Arduino can do it and (b) how to do it.
Also there is the issue of powering the Arduino, is it possible to have an independent power source that is rechargeable or use very small power source to do this (something like a battery that will last for years)?
Anyone got any ideas?
Thanks for any suggestions!

here's my quick take on this. Use the laptops Wake-On-Lan function or Wake-on-Ring function to turn the laptop on externally. To sense the power level, step down( w/transformer ) and rectify the AC power to a safe level for Arduino to sense on the analog input. Power Arduino off some AA batteries and use a interrupt timer to wake it up every hour to check on the AC power situation and reset or increment a counter in EEPROM memory based on what is found. When you see the counter == to 24, turn on the laptop. The Arduino might monitor a USB port power pin or something like that to know when the laptop is running.

that should get you started.

That is an excellent idea. However, my hardware skills are very basic.
So I need:

  1. a transformer that takes 220V AC to 5v DC (40mA)
  2. A self sufficient arduino board (Like this: http://www.instructables.com/id/Self-Sufficient-Arduino-Board/ )
  3. Some sort of device that physically turns on a computer when current is passed through it. Any suggestions?
    Aside from this, I'm confident I can code the logic.
    Also I was wondering if the Arduino board(Duemilanove or Diecimila)'s USB connection can be used as a "sensor" in this case?
    Thanks again for the suggestion.