I'm very new at Linux. I'm running a mediaserver (pms) on Ubuntu 18.04.3 mini, with no desktop or something like that.
Now I'm wondering if it's possible to connect to the server, and run applications like luckybackup GUI, or is it a requirement to have a desktop installed on the server?
What you're looking for is called X-forwarding and it can be done. The server will require to have the X11 server-side stuff installed, and SSH will require to be configured to allow it. Have a look here and see if this helps: https://www.addictivetips.com/ubuntu-linux-tips/set-up-x11-forwarding-on-linux/
If you need cross-platform compatibility then Spice project is a bit more involved but might also help: https://www.spice-space.org/
VNC is another cross platform solution. Here's one tutorial but there are probably plenty more out there: https://www.digitalocean.com/community/tutorials/how-to-install-and-configure-vnc-on-ubuntu-18-04