Text mode browser
You could try a text mode browser, if you have one installed: elinks, links2, w3m, lynx (roughly in order of preference for your use case). They may or may not work depending on whether the login page requires fancy stuff that these browsers don't support.
Download the files on another computer
Since you seem to have access to a computer, you could download the packages on the other computer and copy them. The following command displays the URLs of the packages required to install unity; the second one displays just the URLs with no extra noise.
apt-get --no-download --print-uris install unity
apt-get -y --no-download --print-uris install bup | sed -n "s/^'//; s/'.*//p"
Another approach that leverages another computer would be to use that computer as a router. If you have a smartphone and a USB cable, activate your phone's tethering mode.
Running a GUI anyway
At the GUI login prompt, click on the icon with the Ubuntu logo and you'll see a list of possible sessions (corresponding to the window managers and desktop environments that you have installed). The menu may be accessed through a different UI element if the interface you see isn't the default display manager from 12.04. If you have a session type other than Unity, select it.
If you have no window manager at all from your system, you can still run GUI applications: all it takes is the X server. From a text console, run the following command:
startx firefox -- :1
Replace firefox by your favorite browser. You may prefer to run a terminal emulator where you can more easily start other commands. The -- :1 part instructs X to use a different display number from the default (0), in case there is already a GUI login prompt occupying display 0.
You may even run the X server directly, this has minimal dependencies. You can start a GUI application by specifying the display number through the DISPLAY environment variable.
X :1 &
DISPLAY=:1 firefox
To kill the X server, run jobs -l in the shell, note the job number (which will be 1 if you hadn't started another background job before), and run kill %1 (or whatever the job number is).
Without a window manager, with just the X server, you get an extremely crude interface. There are no menus, workspaces, window borders, keyboard shortcuts or anything convenient. Worse, the only way to switch the focus between windows is to move the mouse cursor, and there is no way to move a window or raise a window that's hidden behind another one. Nevertheless this is a convenient method when you need to run one application (such as a web browser with support for fancy Javascript) in an emergency.
