Or would a Crown Prince typically be educated at court?
What would his education curriculum be?
In the 16th century Wittenberg was the epicentre of Lutheranism, and the Danish Reformation (1536) was for decades directly influenced by Luther and Melanchton, with most of the prominent Danish clergymen having attended the University at Wittenberg.
No Danish crown prince did attend the university at Wittenberg though, and while a "Grand Tour" of Europe was typical of a Renaissance Crown Prince, it can't be said to have been considered typical that he would attend Wittenberg. But I think that is exactly Shakespeares point, to stress the brooding, one can almost say clerical personality of Hamlet as opposed to the average crown prince.
Was "Hamlet" set contemporaneously?