Updating Plex on Ubuntu 16.04 Server

Get the link for the latest version of Plex from the download page. (Look for the Linux link, find the correct Ubuntu version and right-click > Copy Link Address)

Download the latest version of Plex using wget (and paste the link into the terminal after the wget command)

wget https://downloads.plex.tv/plex-media-server/

Install the package using the Debian Package installer

sudo dpkg -i plexmediaserver_1.3.4.3285-b46e0ea_amd64.deb

This should update Plex, you may need to reboot your server afterwards. (sudo reboot now). Reloading the Plex page should show everything up to date.