Is there a way to pipe audio between these two networked computers?

PulseAudio is the sound daemon for the default Ubuntu desktop. There are Windows binaries available.

I also spotted an article from '08, "Streaming Audio from Windows to PulseAudio Server". The Windows PulseAudio builds weren't stable enough for that author, so he went another route, using something called LiveInCode and netcat or Cygwin's SSH as the transport layer. The author used a commercial software called Virtual Audio Cable to create a "virtual audio card", and uses that as the device to which his media player plays, and from which LiveInCode is capturing.

• Server side configuration: add this to PulseAudio's startup options:
  -L "module-simple-protocol-tcp port=4712 rate=44100 format=s16le channels=2"

• Client: LiveInCode via Netcat (susceptible to dying due to closed connection due to a bug in module-simple-protocol-tcp):
  linco.exe -B 16 -C 2 -R 44100 | nc.exe <host> 4712

• Client: LiveInCode via SSH:
  linco.exe -B 16 -C 2 -R 44100 | ssh.exe <user@host> "cat - | pacat --playback"

I've not tested this solution, but it sounds reasonable. Note that more recent versions of PulseAudio may differ in some ways. Check your documentation.

I don't know why you would need to do this, but you don't need any thing special as long as the audio out is working on your old machine and the audio in (mic) and audio out is working on your new machine. You can plug one end of an 1/8" audio cord into the green audio out port of your old machine and plug the other end into the mic port of your new machine, then configure the mic input of the new machine to be out put through the audio out port. I have done this with vcrs and dvd players, should work the same with your old computer.