The start problem may be a bad power button board or a bad power supply.
The only way to test the button board is to replace it.
The PSU can be tested with my guide:
As for the Leopard update, download the update from Apple, don't use the Software Update.
Download this file to your desktop and then update your OS:
Thanks a lot Chris. I wrote this specially to send you by email using the mail app, but decided to post here. I'll buy a multimeter and test it following you guide. Well, this machine is so elegant and I'm hope that my PS won't tease me. My Mac is running for three days without problems, so maybe the mea culpa may be of the button board... like the seller told me days ago.
Also, I forgot to tell you (my tutorial tells you) but you need to be sure and replace the PRAM battery.
A bad battery can cause start problems.
Replace the PRAM battery and reset the PMU.
If you still have problems, test the PSU then replace the button board.
Japamac: I discovered that my problem is the battery because if a keep plugged the cable force after shutting down the Mac, BUT don't turning off my nobreak. So in the next morning I just need to push the start button to enjoying the even strong PM! I read something about this in a discussion group about Macs here in Brazil. Even if your battery have no power, keeping the cable force plugged, this maintain a little charge sufficient to keep the PMU and so on...