Game does not work by default. It launches an obsolete ubisoft launcher that failed to update to latest revision. I installed the launcher using the following instruction https://www.gamingonlinux.com/2022/03/ubisoft-connect-on-steam-deck-guide-with-sd-card-access/ and then just copy/paste an entire folder to drive_c where the prince of persia is installed. so just

1. switch to desktop mode.
2. go to the link above and follow Text Guide: step 1-5
3. find an installation folder for ubisoft connect described in step 6
4. copy the folder and paste (replace) it to /home/deck/.steam/root/steamapps/compatdata/33320/pfx/drive_c/Program Files (x86)/Ubisoft/
5. launch the game.