As far as I know, there is no advanced booking. You just show up and wait in line (and wait and wait and wait and wait).
I've only been there once, and that was in 2005, so things may have changed, but....
I was there on a Friday during the off-season. Got to the MK gates early and was one of the first people through. Still found a 45-minute line. A guy in front of me mentioned that he had been coming to the barber shop there every week for thirty years. The barbers had a lot of fun with the kids they were cutting, and did their best to put them at ease. One little girl was having the time of her life! The barbershop quartet wasn't scheduled to perform that day, but all four of the guys happened to be there at the time and they burst into song. The crowd went wile.
My haircut proper was quite possibly the best haircut I've ever had. The barber himself was kind of a downer; he asked me what I did at home, I told him I was a social worker, and he got all misty-eyed and mentioned how social workers saved his life when he was a kid. He was kind of sad and wistful for the rest of the haircut.