You are aware that you can have a character be a prince and act like a "regular" person, right? "royal" or "prince" isn't a personality, it's just a societal status. sometimes it's not even a financial status, if the kingdom is poor, or at war.
that being said, do you want him to have been raised by the king or not? if you just want the king to be his dad he could have been raised by someone else for a number of reasons (his safety, he ran away, he was kidnapped etc.)
But remember that rich is NOT a personality. Both of the main characters of my novel are rich, one was born rich, one became rich - they still have personalities that have 0% to do with being rich. Your social status has nothing to do with who you are. You can be rich and humble, or poor and conceited or proud. Just write the character as you imagine him.