As stated, this is the vacuum line that operates the heater water shut-off valve, visible in your first photo. I belive it is actually controlled by a lever acting on a valve located behind the glovebox, which is in turn controled by the heater control levers on the console. The theory is that when the "heat" lever is set to full cold, the valve is closed, shutting off hot water.
The best way to repair it is to "sheath" with something like the clear flexible piping that places like Robert Dyas used to sell. I have done this with ours, for the entire exposed length. I used some silicone sealant to seal each end of the sheathing and it works well.
As your shut-off valve looks original, you may find it is now non-functioning - I replaced ours with a newer design many years ago.
The broken hose will not affect AC operation by the way. That probaly needs new gas at the very least? Then you will have to get into the whole R12/R134 "thing"!