Using different computer to download podcast

Yeah, you cannot have your iPod synced to 2 different computers..
Easiest thing I would do is just download the podcast to your own computer.
If for some reason you can't, you could plug your ipod into your friends computer, make sure that disk use is enabled (which will make it appear as a drive to the computer), and then just drag the mp3 onto the drive. When you get home, open that mp3 in your own itunes and have it sync it to the ipod.
There are many other options as far as just transferring the data, like using a flash drive or burning it to cd, so just do whatever is easiest.