apt-offline is really really simple.
1. On your disconnected computer, instead of upgrading by running
"sudo apt-get update" and "sudo apt-get upgrade", run
"apt-offline set --update apt-offline.sig"
2. Then, you put that "apt-offline.sig" file onto a flash drive, and
go to a computer that has an Internet connection, put the flash
drive in and run "apt-offline get --bundle bundle.zip
apt-offline.sig"
3. Now copy that "bundle.zip" file to your flash drive and go back to
your disconnected computer. Run "sudo apt-offline install bundle.zip"
4. TADA! Your disconnected computer has been updated.
Of course you need apt-offline installed on both computers, but
according to this chap you should be able to copy just the apt-offline
deb file and install it the "dangerous" way using "dpkg".
Sorry, just one small correction. Do the steps as I mentioned, and that
will update your repository indexes. To upgrade your packages, follow
those steps *again*, but this time use:
$ apt-offline set --update --upgrade apt-offline.sig
Remember that you need to use the first command the first time,
otherwise your computer is not going to know which packages to update.
You could probably even make it a bit easier by executing the commands from the mounted flash drive each time. That way the files will be created on the flash drive and you won't have to copy the files manually.