How do I upgrade releases from a Live USB with the command line?

Sure. It's fairly similar to most remote-fixes (namely fixing Grub) where you chroot into the system but mount some the runtime elements from the Live environment over the top. It's quite easy but it sounds and looks horrific if you're not used to it!

The following assumes that / is the live environment and that sdb1 is the disk where the actual install is (check that first with sudo fdisk -l!). Other than that, it's just a pile of commands to throw in:

sudo mount /dev/sdb1 /mnt
for i in /sys /proc /run /dev; do sudo mount --bind "$i" "/mnt$i"; done
sudo chroot /mnt

By this point you're root on your actual system and you can fix a whole load of things. Obviously this is the point where you need to be careful what you do but as long as you're not running around rming a load of stuff, you'll probably be fine.

You probably could run the graphical updater but why muddy the waters with silly Xauth issues? The following will update your install's apt databases, download the command-line updater (if it's not installed already) and run it. You're already root so sudo isn't necessary but I'm including it just in case you've dropped to another user (it doesn't hurt)

sudo apt-get update
sudo apt-get install ubuntu-release-upgrader-core
sudo do-release-upgrade

When you're done, just reboot onto your main disk.