You will absolutely have to go land-side.
Terminal A is basically 3 terminals in one - none of which are connected air-side. The United shuttle goes only to the United part of the terminal where United (Express) flights depart from. Given you've said you're not flying United, that'll obviously be the wrong part of the terminal.
The bus can also be somewhat unreliable, so you'll be best to avoid it entirely. From terminal B you will need to go airside, catch the free AirTrain to terminal A, and then re-clear security in whichever of the security checkpoints leads to your gate.
Whether this is possible in 65 minutes will depend mainly on how busy security is at the time, and whether you have TSA PreCheck and/or priority at security. At best, it's going to be tight. If you've got checked bags and those bags are not able to to checked through to your final destination then I'd say it's impossible.