What are the best practices for updating and maintaining the Whonix Gateway? How often should I run apt-get update and apt-get upgrade?
You should update both Whonix-Workstation and Whonix-Gateway at least once a day
At least once a day users should update the system package lists  with the latest version information on new and updated packages that are available for download
Also keep an eye on our forums in case there is a security problem in apt/Tor/the kernel that may need special/urgent upgrade instructions.
What about how to do the upgrade itself? Can I keep using the workstation while apt-get upgrade is working?
I’m not sure I understand your question. Do you mean just running
sudo apt-get dist-upgrade by itself?
Of course. Waiting for
apt to finish is not necessary.