Which preposition is used : to beat someone at/in a game?
Also when is it used ‘you beat me on this’?
As is often the case with questions about prepositions, the answer is probably more complicated and idiomatic than you would like. Here are some thoughts that come to mind.
If you are talking in general terms about a certain type of game, then you usually use “at”: “I always beat him at chess”, “When were you last beaten at tennis?”
You can use “in” when talking about particular contest: “We were beaten in this afternoon’s game”.
You may hear “on” used with certain games that you are perceived to play “on”, such as video games: “He beat me on Space Invaders” (sorry, I don’t know any modern computer games!) In these cases I think you can always also use “at”, and “at” sounds better in a formal setting.
There are probably other considerations that I have not mentioned.