I want to fix the Shellshock issue, and the usual solution is to update Ubuntu.
How can I update only Bash and without having to reboot?
You don't have to restart for the Bash update to take. Just update in the normal way.
sudo apt-get update sudo apt-get dist-upgrade
If you (for whatever nonsensical reason) need to limit that to Bash, you can:
sudo apt-get update sudo apt-get install bash
And again, because of the way this exploit is launched (setting an environment variable before starting a new instance of Bash), you don't need to reboot after you do this.