Ouch… that’s really old.
You should be able to simply download the current version from the plex homepage and put it in your /Applications/ folder:
https://www.plex.tv/media-server-downloads/
PMS version 1.19.3 needs your macOS to be on Mavericks at least (macOS 10.9).
Best make a backup of the app and the Plex Media Server data folder, preferences (~/Library/Application Support/Plex Media Server/, ~/Library/Preferences/com.plexapp.plexmediaserver.plist)… just to be safe.
https://support.plex.tv/articles/201539237-backing-up-plex-media-server-data/
