If you go to: https://mullvad.net/en/download/vpn/linux
And click the "download .deb" button (It says underneath "Works on Ubuntu 20.04+, Debian 11+ (64bit only)". As long as your Ubuntu is up-to-date, this will work fine)
you get a file ("MullvadVPN-2023.6_amd64.deb") you can run just like on Windows (similar to MullvadVPN-2023.6.exe)
opening the file should open a GUI for installing the file
Keep in mind, to update Mullvad VPN, you would need to download a newer .deb file (after an update is released). It shows the latest version above the download buttons, below the "Mullvad VPN for Linux text" This is the same as how it is on Windows
Edit: This is not intended as good advice, just a simple way to install Mullvad VPN. The smartest solution would be to add the repo.
2nd Edit: While this is how Mullvad provides their software, it is never ideal to install random .deb packages or add third party repos without being sure that the ones who provided the package/repo is trustworthy.