You do not need the Internet on the computer you are installing Ubuntu on; however, you will need the Internet to get Ubuntu. You will need a blank DVD or a USB drive to create an install disk. There are some excellent instructions on the Ubuntu website. Also, does the "other PC" have a hard drive? The disk size for installing a Linux-based operating system is fairly small and a 1.5 TB hard drive seems a bit overkill. You can install Ubuntu to the 1.5TB hard drive on the "other PC", but I wouldn't install it on the "decent PC". The potential to hose your working PC is too great if you are not experienced. So here are some steps to follow:
1. Create an Ubuntu install disk on your "decent PC".
2. For best results, if your "other PC" has a hard drive, use that disk to install Ubuntu on to.
3. If the "other PC" does not have a hard drive, yes, you can install it on the external drive, but, depending on the purpose of the "other PC" it might be overkill, and, depending on the type of USB, you might get a performance hit.
To install updates and proprietary files during the install have you considered hooking the "other PC" in place of the "decent PC" for a short while?