"You must be logged into Steam to play Batman: Arkham Asylum" error message

You not only need to be logged into Steam, you also need to buy a copy of Batman Arkham Asylum and add the game to your Steam account. Instead you've used the "Add Non-Steam Game to My Library" option to add a shortcut to your library. This doesn't add the game to your Steam library, it only lets you launch games that don't require Steam from the Steam client. It doesn't work with games that require Steam, like Batman Arkham Asylum.

Once you add the game to your account and download it, it will look like this in your library: