Because the Duke of Sussex is not the eldest son of Prince Charles, his son (Archie) isn't a British Prince or even a Royal Highness.
Archie is, however, seventh in the line of succession to the throne.
Since his mother, Duchess Meghan, is not of royal blood and not British, he is the first half American and half British member of the British Royal Family in history. Archie Harrison is also biracial due to his mothers heritage.
He was christened in a private ceremony on July 6th, 2019. At this point, we do not know who his godparents are.