This is an interesting question because the reader must make inferences from the text in order to draw any conclusions on this topic. It would probably be helpful to start with brainstorming what kinds of privileges a prince would have in any kingdom and then see if we can find or infer evidence of those in the text. Princes would usually benefit from:
1. access to the king and queen
2. deference from servants; influence over his "lessors"
3. deference from people of lesser status, even important people of the court who aren't royalty
4. freedom to do as he pleases regardless what others think
I would suggest that we see all of these in Shakespeare's play. Hamlet clearly has access to the king and queen. They are clearly concerned with his mournful behavior and he has an opportunity in Act 1 to explain himself, telling his mother that no matter how he acts on the outside he "has that within which passeth show." He is tyring very hard to give her a sincere response to their concerns. In Act 3 Hamlet goes to his mother's rooms and speaks very harshly to her about her inappropriate marriage to Claudius. He speaks with no censor on his thoughts; no one else would be allowed to speak to her that way, especially on that topic.
We see deference from servants in Act 1 when Hamlet wants to go to the castle top to confront the ghost. All of the servants and his friend Horatio warn Hamlet of the danger of such an act, but he tells them he will not be stopped and threatens "I'll make a ghost of him" that tries to stop him. They all back down and leave him to his own devices, but follow him at a distance to be there if needed.
We also see deference from Polonius, an important member of the court and Claudius's "right hand man", in Act 2. Hamlet is putting on his crazy act and seems to not know who Polonius is. Instead of getting upset with Hamlet, he cautiously engages in a conversation with him. Hamlet calls him a fishmonger. Polonius merely responds, "Not I, my lord" rather than antagonizing him with a longer explanation. When Hamlet says something about Polonius being an "honest man" Polonius doesn't really know what to make of the comment, but instead of pressing the issue, he responds by repeating the comment and agreeing with Hamlet even though it is doubtful that he truly understands what Hamlet is really talking about.
Even though Claudius sees Hamlet as a threat to his own safety and kingship, he doesn't have that much control over him. He tries to bring in friends (Rosencrantz and Guildenstern) to help control him and to serve as spies, but Hamlet soundly thwarts their efforts. He tries to get Gertrude to calm him down, but she can't and is actually turned against Claudius to some degree in the process. He tries to send Hamlet to England to be executed, but Hamlet is able to use his status as Prince to convince pirates to return him to Denmark and deliver "threatening" letters for him. Claudius and Laertes are a bit "at his mercy" until the final show-down in Act 5 scene 2. In Act 4 Laertes claims that he "will not be juggled with," but it is clearly Hamlet who can rightly make this claim.