This is one of those questions that has two answers - the serious and the fanciful.
In the case of a fairytale like Rapunzel, you can safely ignore the royalblood's lack of training in favor of the mystery and "magic" of a fairytale. It's fiction!
In a more serious novel, however, you set yourself ahead by thinking of realism aspects like this.
There are several ways for the newly discovered prince or princess to learn the ways of the kingdom. As Airianna noted, Rapunzel was able to learn under her parents' direction. A mentor could fulfill a similar function, like Josiah was guided by the high priest of his time. Tutoring is a good solution as someone else mentioned. And then there is the art of trial and error!
I think you also need to consider the political structure of the country at hand. How much responsibility does the king have? Some royalty are just figureheads and have little political function, so training is perhaps academic. Other nobles have a lot of advisers that could assist them.
In Peter's Angel, which will be an alternate history colonial novel (Lord willing), Peter's father was related to a governor of a colony. So while he was comfortably middle-class, his children were raised more as commoners than royalty. So when death puts Peter's father abruptly in control of a colony, it's a shock to Peter - but he uses his common upbringing for good to be level-headed and down-to-earth, like Michael suggested.
On the contrary, Nathan was born royalty and fled from it. I'm toying around with the idea of raising him as a nobleman's son as opposed to a peasant. Part of the reason for that was to give Nathan more experience with politics. Regardless, his adoptive parents have been preparing him, at least emotionally, for the fact that he should one day retake his throne. So he knows more about royal life, particularly in the form of memories from the years before his parents were killed, than he's willing to admit.
) or prime ministers.
