I assume that those two computers are located next to each other because otherwise this information would be meaning less.
Now to install packages on Ubuntu system, you have two options:
Share the Windows PC's internet connection with the one running Ubuntu. For this connect the two computers using a direct network cable or by using a router if you have one. Then setup you windows PC to share internet connection and change Ubuntu PC's network connection properties to assign it a manual IP and to use the Windows PC's ip as gateway. If you decide to use this method then I can provide more detailed info.
The second method would be to download packages and install them offline. There is a very good answer by Javier Rivera here, I am just copying it here for completeness sake:
"Check out Keryx, it's an offline repository manager- it lets you download updates and new programs (with dependencies) to your flash drive. Its interface is similar to synaptic but it works on a pendrive (don't need installation). Unfortunately, the GUI needs wxwidgets, which don't come preinstalled on ubuntu (they're cross-platform and installable from here and ubuntu repository here). It can only install software in a Ubuntu system, but you can download the updates or new packages in any Linux, Windows or OS/X.
Here you can find a tutorial."