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/1.3.4.3285-b46e0ea/plexmediaserver_1.3.4.3285-b46e0ea_amd64.deb

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.