Plex should be just fine with moving the drive.
I could not say one way or the other how well ubuntu itself will behave.
I would assume it should boot, but may need some updated and/or different drivers for the new motherboard/cpu/other components.
Personally, I would look at it as an opportunity to ‘clean house’ and start with a fresh ubuntu and plex install.
You can keep your old server up and running while getting the new one set up the way you want.
