would know his "soul" better than his priest?

I assume that Montresor is not speaking his story aloud to anyone, including any confessor. The story looks like something that was written out on paper. There are too many details which a person speaking directly to another person would not be likely to include, especially fifty years after the fact. To take one of many possible example:

“Let us go, nevertheless. The cold is merely nothing. Amontillado! You have been imposed upon. And as for Luchesi, he cannot distinguish Sherry from Amontillado.”

How could Montresor quote Fortunato verbatim if he were speaking to another person extemporaneously? 

I believe that Poe intended to have the reader assume that he, Edgar Allan Poe, had come into possession of a document, probably an old letter, which he had translated from a foreign language, most likely Italian, but possibly French or Latin, and was now publishing in a magazine. I assume that Montresor wrote a letter to a person whom he calls "You, who so well know the nature of my soul," and that the recipient kept the letter which was found among his papers when he died. Both Montresor and his confidant, or confidante, must have been quite old and close to death. Alternatively, Montresor might have written the letter one night when he was drunk but decided not to send it the next day when he was sober. (Some of us have done that ourselves, haven't we?) In that case the letter might have been found among Montresor's own papers when he died.

I do not believe that we are supposed to be listening to a long dramatic monologue, as is the case with Robert Browning's "My Last Duchess" and other such Browning poems, but are reading something that was putatively written by Montresor himself and then somehow came into the possession of an American editor named Edgar Allan Poe, who naturally would have had to translate it into English before he published it. Edgar Allan Poe may put his name on the piece because he is the owner, the translator, and the publisher--but not the author! That is Poe's fiction. The story is related to Poe's "Ms. Found in a Bottle." Edgar Allan Poe supposedly did not write the story: he found it in a bottle--or else somebody else found it in a bottle and passed it on to Poe.

Montresor would have been around forty when he murdered Fortunato. He says that fifty years have passed since that event. That would mean that he would be ninety when he told the story. He would probably be dead by the time it was published. If so, he could hardly be speaking it aloud--unless he is dead and is confessing to St. Peter!

I believe that Montresor supposedly wrote the confidential confession one night and that it survived on paper until it fell into the hands of Edgar Allan Poe. It was intended "for the eyes only" of one person. This technique makes it possible for the author to leave out a great deal of exposition. For example, there is no need for the author (Montresor/Poe) to say that the story is set in Venice. "You, who so well know the nature of my soul" knows where Montresor lives. I imagine the confidant, or confidante, to be living in France, since Montresor would be more likely to send a letter to France than to someone living nearby in Italy. That is the simplest explanation.

Many people have speculated about this question. It is fairly common to read the assumption that, because of his age and concern about his afterlife, Montresor is making a confession to a priest. To me, the narrative does not sound like a verbal communication to a second person who is present in the same room. It sounds like a written manuscript. There is no suggestion in the story that anyone else is present when Montresor reveals what happened fifty years ago.

Some examples of long verbal narratives to a live listener are to be found in the Sherlock Holmes stories. Sir Arthur Conan Doyle always takes care to make it clear that the listener or listeners, Sherlock Holmes and Dr. Watson, are present and taking a keen interest in what they are hearing. For example, in "The Adventure of the Speckled Band," Sherlock Holmes keeps making comments and asking questions. Otherwise the back-story being told by the client Helen Stoner would turn into a story in itself rather than being only part of an interview. Here are some of Holmes' interjections:

“I am all attention, madam.”
Holmes nodded his head. “The name is familiar to me,” said he.
“Your sister is dead, then?”
“Pray be precise as to details,” said he.

The same sorts of comments, questions, etc., can be seen when Holmes is listening to the long back-story of Jabez Wilson in "The Red-Headed League" and in other Sherlock Holmes stories. Poe could have done the same thing if he had wanted the reader to think that his story was a verbal communication or confession. But there are too many specific details which it seems unlikely that a person would remember if he were only speaking in confidence to a friend or a priest. For instance, Montresor tells the names of the two bottles of wine he shares with Fortunato. There are also too many places where a listener might want to ask a question and no question is asked.

In my opinion, the story is intended to be taken as a confidential letter from Montresor to a man or woman he has known for over fifty years. It would have been written in Italian, French, or possibly even in Latin, and found among the papers of the confidant or confidante after his or her death. Or else it might have been found among Montresor's own papers when he died, because he wrote the letter one night while drunk on French wine and decided not to send it next morning when he was sober and thought better of it. (Many of us have done something like that at one time or another, haven't we?) The document would have somehow gotten into the hands of an American editor named Edgar Allan Poe, who translated it into English and published it in an American magazine.

This theory cannot be proved, but it seems the most likely. Poe did something similar with his story "Ms. Found in a Bottle." Someone wrote a long narrative, sealed it in an empty bottle, and tossed it overboard just before his ship was sucked into a whirlpool. Edgar Allan Poe got possession of the manuscript and published it. Poe was an editor himself for many years, and he probably thought like an editor.

Poe pretends that Montresor is communicating with somebody he calls "You, who so well know the nature of my soul" because this enables Poe to leave out a lot of exposition and to focus on what is dramatic. Presumably this confidant or confidante knows a great deal about Montresor's past history and present condition, including such minor details such as what city he lives in. The man or woman to whom Montresor was writing may have known all about the thousand injuries of Fortunato, so that these would not have to be described either.