Hi, Niels, and welcome ~
Most AN surgery patients do experience some improvement in their ability to balance, following surgery. It depends on how much the vestibular nerve has been compromised by the tumor prior to surgery and if the brain has compensated for the loss of input. If the nerve is still partially operative, the brain will receive confusing signals and that will precipitate a loss of equilibrium. Following AN removal surgery, the patient will need to undergo some balance therapy in order to re-gain a relatively normal sense of balance, although it won't be exactly as it once was (pre-AN). In some patients (I was one of them) the vestibular nerve has been so badly damaged by the growing tumor that the brain stopped receiving signals some time ago and has already (mostly) compensated, so re-gaining balance is fairly simple and rapid. Again, my balance is not 100% of what it was prior to my AN but it is quite serviceable. From the information you offered, I'm guessing that you should re-gain most of your balance ability fairly soon. Running shouldn't pose too much of a problem, once you get back into it. Some of our members that underwent AN removal surgery have run marathons within a year of their surgery. I'm not a doctor and can't offer any guarantees but with some luck, determination and practice, you should be back to your running regimen fairly soon.
Jim