Okay I got it working, I've had this issue for a couple years and just accepted I'd only save progress on my desktop but I finally figured it out:
- Sign out of your account on the Xbox app and close it.
- In Sudoku, click your profile at the top right to open the Xbox app, and sign in. (Simply opening the Xbox app and signing in didn't seem to work for me, I had to click the profile in Sudoku, which opens the Xbox app to sign in. Maybe something about Sudoku looking for the sign-in only if it was the app that prompted it).
- After signing in, go back to Sudoku, it may prompt you to sync, which should then show your achievements.
- My monthly rewards were still not synced at this point. From here I simply had to close Sudoku and re-open it for them to pop up.
This is ROUGHLY what worked for me, I did it across three devices (Windows 10 Desktop, Windows 10 Laptop, and Windows 11 Laptop) and it was a bit finicky, taking a slight variation on the above (whether Sudoku app was open or not during Xbox app steps, etc.) and a few tries at times. I can confirm that all 3 devices now show the same level, achievements/awards, and monthly rewards/medals.