You can stop `apt-get` temporarily with `Ctrl-Z` and then restart the job in the background with the `bg` command.
$ apt-get foo bar ...
^Z
bash: suspended apt-get
$ bg
bash: continued apt-get
$
However, when you do you'll continue getting the output from `apt-get` on the terminal. It _isn't_ possible to stop that after you've already started the command (other than with dirty hacks using a debugger). You can run other commands here, though. While the job is stopped, it is still alive but not running, so it won't make any progress but you won't see any output either.
What you may find useful is the `screen` or `tmux` commands. They let you run multiple sessions with different terminals within the same physical terminal. If you're using X, you can run additional terminal emulators or perhaps create a new tab in your current one. Whichever way, you can run `apt-get update` in one shell and then switch to another to continue working.