The narrator conflates his religious/spiritual side with his romantic affection for Mangan's sister. With this combination, his ideas about her and his role rise to idealistic heights. He allows himself to get caught up in this increasingly majestic world of himself as a knight seeking treasure for the object of his affection:
Her name sprang to my lips at moments in strange prayers and praises which I myself did not understand. My eyes were often full of tears (I could not tell why) and at times a flood from my heart seemed to pour itself out into my bosom.
When he completes his journey and arrives at the bazaar, he is heartbroken that the exotic-sounding "Araby" is simply an English profit machine. Recall that he would watch Mangan's sister defined by the light of the open door. As he leaves the bazaar, he is in darkness. He realizes his majestic quest was nothing more than an errand to buy a useless souvenir. He has an epiphany that his majestic quest was all an illusion. He is mostly angry at himself for putting too much of his heart and mind into this illusion.
The ending of this story is closed in the sense that we know the narrator has come to a new realization. He was full of idealism in the beginning and is frustrated with anguish in the end. The only way we can say this is an open ending is if the reader were to speculate upon what the narrator will do with this epiphany. Will he avoid Mangan's sister or will he try to reconstruct his dream of her in more practical ways? This openness is all speculation. So, it is more accurate to call it a closed ending. The story ends with the closure (closed) that the narrator has a revelation, even if the revelation is pessimistic and dark in tone.