In this story, we are not directly told how John feels about being a priest. He does not come out and explicitly tell us how he feels. Therefore, we have to try to read between the lines of what he does say so that we can determine how he feels. I would argue that John has felt at times that being a priest was a burden but that, overall, he feels that it is a privilege.
At the beginning of the story, John tells of his early training. When he does so, it sounds as if it was something of a burden to be trained as a priest. He tells us that his father went into the house of an ancient dead person to get metal and that he, John,
… stood by the door and my heart felt small and weak. … But it is not fitting that a priest's son should show fear.
This seems to indicate that learning to be a priest is something of a burden.
But later on, when John is talking about things that happen in the “present,” he is clearly proud of his status as a priest. He repeats “I am a priest and the son of a priest” as something of a mantra. For example, at one point he is about to be swept out to sea. He tells us
I grew angry then—my heart felt strong. I said aloud, "I am a priest and the son of a priest!" The gods heard me—they showed me how to paddle with the pole on one side of the raft. The current changed itself—I drew near to the Place of the Gods.
From this, we can see that he feels that being a priest is something to be proud of and something that makes him special.
Thus, we can say that John once felt that being a priest (or at least training to be one) was a burden but that “now” he feels that it is a privilege.