If I have a directory which contains a bare filesystem(ie bin, dev, home, lib, opt, media, etc) but without any of the drivers or programs or anything on it you might expect, how do you go about installing those things? I can't just use apt-get or whatever, because that would just put it on my main filesystem, not this mounted one. Is there a way to do this kind of thing?
You can install from source and tell the system to compile the software with a specific path. Biggest issue: dependencies ... you need to sort those out yourself. And install them from source too.
I would advice to use a source-based distribution rather than Ubuntu if you want to go that path. Like Gentoo, LFS, Crux.