Yes, until someone with the authority to tell him he's not eligible tells him he's not eligible.
He's no less eligible than Ted Cruz (born in Canada to an American mother and Cuban father). Of course, Archie would have no renounce any royal titles. A minor formality.
The problem with Natural Born Citizenship as defined in the Constitution is that there seems to be no enforcement mechanism other than the will of the voters and if Archie decides to run for president and actually wins the election, who's going to stop him from assuming office?
Hillary.......its her turn.