My son is 9 now and we are at last able to go to the barbers with minimal stress.
First of all, I go at the quiestest time possible - the barber near us is very quiet on a Monday afternoon - usually there is no queue at all or maybe just one other customer. Its quiet and calm. Ask your barber when their quietest time is. You could always ask them to turn down any music playing.
Try out a few barbers until you find a place that is understanding. I went to a few places that didn’t seem to have ever heard of autism - but many are used to children with autism.
I have a quiet word with the barber before each haircut to explain he has autism and how to manage him.
I ask them not to try to chat with him or make jokes as it will just annoy him while he feels stressed - just keep talking to the minimum needed to get the haircut done.
No paper around his neck, no spraying water directly onto his head. They spray it on the comb away from him.
He holds a cloth over his mouth to stop hairs going in as that used to upset him.
He brings his tablet - we upload cartoons from netflix and he watches some cartoons on mute - Angry Birds is good as there is no talking in it so it is fine on mute - and it is funny so it distracts him. Being distracted is a big help as after a short while he stops noticing the scissors or the razor.
He gets a small reward afterwards.