Was on the Fantasy, Apr 15 - 23. Even though I was a first-time cruiser and I had to wait till I was 75 days out, I was able to book my five year old grand-daughter an appointment on the day of the Prince/Princess dinner. I seem to remember that there were several time slots available at the time as well. I bought her an entire makeover package including, makeup, hair, a Jasmine dress, and shoes. This cost about $240 and it was well worth it. The stylists call themselves Fairy Godmothers in Training. Ours engaged our grand-daughter, Savanna, with little stories; she explained what she doing at every step, and made sure that Savanna was always comfortable.
I do not think you will have any trouble booking an appointment. Even if you have to wait till you get on board, I do not think you will have any problems. Our appointment was at 12:45 and when we arrived, there was only one other client inside. Slowly more and more clients came in and they were full at about 1:15. You probably could easily have secured a 12:00 appointment on that day.