Machine shuts down, might start again in a day or so

Boot up using the osx disk.

DriverPack Solution driver update software Download Windows 7 XP 10 8 and...

It sounds like that key is sticking.

You might want to look on eBay for a replacement keyboard. I'm not sure how difficult it would be to unplug the old keyboard and plug in the new. Opening the iBooks is a challenge, but maybe the keyboard can be plugged in through the hole in the top. Not sure.

Good luck!
Please rate me if this helps.
Bot

Shut down the computer and start up whilst holding down the mouse button. This may take some time, but keep your finger on the mouse button right up until the disc comes out or the log-in screen has appeared.

Check out the discussion and possible solutions at:
http://macosx.com/forums/mac-os-x-system-mac-software/296304-ibook-g4-sudden-shutdown-when-unplugged.html

Sounds like you have a problem with the logic board. This appears to be a common problem with iBook G4 related to the fit of the logic board (ie motherboard) within the case.If you were to take it to apple store the likely fix would be to replace the entire logic board which is very expensive and probably not fiscally sound on a 5 year old machine. Another option would be to try to fix it yourself. If you are feeling brave you can dissemble the laptop and attempt to shimmy the logic board back into position.

Here's a link that detail the approach. http://www.coreyarnold.org/ibook/. This guy had success on two ibooks that had the same problem and they have been working for 2 years since he did his repair
Also, here is a link to a forum that may offer you a few more approaches to solving your problem. iBook G4 won't boot up - fan running & black screen - AppleNova Forums

Hope this helps, good luck!

have you gone into airport utility?
are you using an airport extreme? if so, reset it.
then delete the current wireless network, and try setting up a new one in airport utility.

Replace the on board capacitor with a Panasonic Capacitor 5.5V 0.22F Model number:EECS0HD224H cost about $3 . You will need to desoldier and soldier the new one on, it's easy even for a novice.

It is a kind of on board battery contrary to what most idiot mac users say as a battery IS a form of Capacitor.

Some computer react this way when the logic board goes bad or have a defect in it.  
First, Try this:  
Remove the battery.  Plug back in the power cord, then press and hold down the power button til you hear a loud beep sound. then release it.  If the computer starts back up... Then you should be okay.
If not, it may has something to do with the logic board.
Let me know if the above works... We may can try some other things if not.
~ enov8

next time you experience this, go to utilities.activity monitor..it will show you what program ir running the "hottest:, and the graph will display what is currently going on. you may have something running your not even aware of...