Switch to desktop mode and enter file explorer. The Nuclear Throne folder should be located at "/home/deck/.steam/steamapps/common/Nuclear Throne/". Paste the mod files as normal, open your steam library, click "Add a Game" on the bottom left and "Add a Non-Steam Game..." Click Browse, and once again head to the Nuclear Throne folder and select the .exe for Nuclear Throne Ultra. Once the mod is in your library, right click it for properties, click compatibility, check "Force the use of a specific Steam Play compatibility tool," and the first option, "Proton Experimental," should work. After that it should run just fine!
To make it look nice, you can add a logo and custom background by downloading PNGs for the Nuclear Throne Ultra logo and some gameplay for the background like one of the photos on this page. Right click the empty background and it should have an option to set custom logo and set custom background.
If you ever need to transfer it, your game data will be stored in "/home/deck/.steam/steamapps/compatdata/[some string of numbers]/pfx/drive_c/users/steamuser/AppData/Local". You'll have to look for the numbers, but I believe it should be one of the longer ones. The "NuclearThroneUltraMod" folder can then be transferred from the steamdeck to a computer or vice versa to transfer the save data.