"See there goes the king's son." this sentence means the king's son is very kind of man for people and he did good to others, so people admired his deed to said "see there goes the king's son(we see that the king's son did good)". Is my understanding right? Yes it is correct because in this case. the king's son is being held up as a role model for others to admire.

No. When people say that, they usually add something specific like "There he goes, doing what I asked him not to" The possible meanings for "There we/you/he/she/they/specific person goes (go if you are using they or we)" are differentiated by someone's tone. If someone says that in an angry tone, they usually mean they don't like what said person is doing. However, if someone says "There you go" in response to someone not knowing what to do or where to go, they mean that the person they are talking to is on the right track. For example, I don't know how to solve this math problem. My teacher shows me how to solve it and I say the steps I think he/she took to solve the problem. "So I carry the 2x over to the other side and divide both sides by that number?" If you are right, they might say "Yep. There you go. It's that easy." As for "See, there goes the king's son," I need to know the conversation, tone, and context. It's hard to determine the meaning when just giving the sentence. It could mean the king's son arrived and he/she is pointing that out because someone doubted that. It could also mean what I described above and some others. There are many ways to use that phrase. I won't go over every single one but there are a lot of ways to use it.

Dennis is right ,as the context will determine the exact meaning of this phrase,since it could have various connotations. The context is as follows: He always stopped to help people who needed it. Soon he made many friends. All the people began to like him. They would say, "See there goes the king's son. He would be just the right kind of king for us." Here it probably means that the person is on the right track (quoting Dennis),it is like an indirect way to vote for him to become a king due to his good deeds.