In chapter 3, "Gotama," Siddhartha and his friend Govinda meet the Buddha. They have just left the samanas in the woods after Siddhartha rants that they have nothing to teach him. He feels disillusioned and has put their master under a spell to release them. Clearly in thinking he knows better than his master in this earlier chapter he is showing his arrogance.
In chapter 3 it is not so clear. Both Siddhartha and Govinda go to meet the Buddha. Both are impressed with this spirital teacher and his teachings. But Siddhartha seeks to question the Buddha. He points out the one flaw in his teachings, the one place where causality seems to break down, ending his question with the comment, "Forgive me if I raise this objection." To me, this seems a sort of false modesty: Siddhartha is being smug, showing off his intelligence and showing that he can find fault with the Buddha's teachings. The Buddha even calls him on it, pointing out that thinking too much about this one objection will be an obstacle for reaching enlightenment. And then he warns Siddhartha: "Be on your guard against too much cleverness," knowing that thinking too much about enlightenment can prevent one from reaching it.
Finally, I think Siddhartha shows he is arrogant in the closing lines of the chapter, where he says that "I have seen one man, one man only .... before whom I must lower my eyes. I will never lower my eyes before any other man." If he thinks so highly of himself that he need never honor another spiritual teacher but the Buddha, that none are worthy of his attention and honor, then yes, I think we can say that Siddhartha is arrogant.
Hesse, Hermann. Siddhartha. Translated by Hilda Rosner. New York: New Directions Publishing, 1953.