Try it. The standard Ubuntu CD can be booted to and you can test it out before you install anything.
The wireless is a Broadcom BCM4318 and that should be supported by the b43 driver but you'll need the
firmware-b43-installer to get it working which will mean you need a cable plugged in for part of the installation process:
sudo apt-get update
sudo apt-get install firmware-b43-installer
Again you can test that in the LiveCD without harming the computer. You'll need to reload the driver though (rebooting isn't an option, for obvious reasons):
sudo modprobe -r b43 bcma
sudo modprobe b43
I do appreciate this is probably all Greek and does nothing to lessen the image that you need to use the command line to deal with Linux, but it's an old computer with only old, ropey drivers available... Sometimes a bit of copying and pasting is a necessary evil.