Can batman arkham asylum run on a old tv like 14 inches

Yes it is possible for a console to be connected to older small screen tvs and still provide a video pf the game being played.
If you want to run the game a game console must be connected to a TV. The TV described will not deliver the same results as a HDTV with a wider screen and may not have the ability to connect to a game console depending on the TV. Buying optional connections is just throwing away your money that should be invested in a better TV even a used one.