There are two problems with drylining:
1) you lose a significant width in your hall, which you will really notice as you carry things in and out (will furniture still fit?). Future purchasers will notice too - it will feel poky
2) it will ruin any original features like cornicing or picture rails in the hall.
I would recommend adding the drylining on the outside in the passage, if your neighbour agrees. Internal plasterboard with a backing insulation will also work, but ultimately it's a all a question of depth and quality of your insulation.
It might be worth approaching your neighbour about adding a door, especially if you volunteer to pay for it and supply free keys. Take the security angle - you're concerned about prowlers breaking in at the back. Also, is the shared passage owned by you or him? One of you must own it, so if it's yours, he only has right of access, and you would be within your rights to add a gate.