Wouldn’t he be an Honorable?
The nittiest of nitpicks: this will not happen if Charles pre-deceases the Queen.
No - they only relate to children of lesser nobility (Barons, Earls and the like). Harry is a Duke - the highest rank of nobility - which means, as first born, Archie would typically hold one of his lesser titles (did someone mention Earl of Dumbarton?) Even younger children of Dukes get to be Lord or Lady x.