A quick search at biblegateway.com reveals only one seemingly relevant passage.
The sentence is written in the reflexive, which seems to indicate to me that someone other than Paul took scissors (or whatever they had in the New Testament) to Paul’s head, rather than Paul doing it himself. The passage does not mention if he paid someone else to do it, or who that someone was, or whatever.
My WAG is that, in Biblical and pre-Biblical times, hair-cutting was something of a religious affair and was thus done by a priestly personage as part of a ceremony. But I could be way off.
Then there’s also the likelihood that the woman of the house was in charge of cutting the hair of her husband and children, leaving me to believe that women probably cut each other’s hair. Again, just a WAG.
I wouldn’t be surprised if the Romans had something akin to professional barbers, as all those busts I’ve seen of Caesar and the like have such nicely trimmed hair. But then again, that brings us back to the whole religious ceremony/woman of the house issue.
So, does history record the first professional barber?
And when were scissors, as we know them, invented?
This site has many interesting details on the history of Barbering, beginning with prehistory and continuing through the practices of the Egyptians, Greeks, Romans and on up to the present day.