Found a reference for a Robert Arnott as an apprentice indenture in 1753 in bugthorpe. Given from my reading there were anly about 40 families living there at the time, it's a good chance this is the Robert of our common line.
With our Robert being born c1737, he would e 16 when he apprentice and about the right age.
Here is the reference:
418 "Jno. Mayor Joyner & Cabinet also burgess of this Corporation produced an Indre of Apprenticeship dated 31 Oct 1753 -- wherein it appeared that Robt. Arnott Son of Elizabeth Arnot of Beverley hath bound himself Appr. unto the [said?] John Mayor for Seven Years Which Indre is Witnessed by Dorothea Mattinson, Suckling Spendlove".This was copied from the Register of Beverley Apprentices in the Beverley Record Office."