You have asked more than one question so I have had to edit it down to one question. Hopefully with the help of the information about the characters you can answer your other question about conflict.
Well, in this story then, the protagonist is clearly the narrator, who is a rather strange protagonist. As he is an unreliable narrator we as the readers are unsure of how much to believe his words and what he says. He repeatedly states that he is sane, but we have deep suspicions that in fact he may not be because of his mannerisms, reasoning and speech. Note how he introduces himself in the first paragraph:
True! - nervous - very, very dreadfully nervous I had been and am; but why will you say that I am mad? The disease had sharpened my senses - not destroyed - not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad? Hearken! and observe how healthily - how calmly I can tell you the whole story.
The juxtaposition of the narrator's insistence that he "heard all things" in heaven and earth and many in hell with his insistence that he is sane, clearly makes us suspect otherwise.
The antagonist is the "Old Man" who the narrator clearly knows well. The narrator says "I loved the old man." The only dislike he has is of the cloudy film that covers one of the old man's eye. According to the narrator, he is apparently rich. Apart from this we are told little of him accept of his horror as he realises that he is going to be killed.
Hope these brief character sketches help you work out the conflict and how it is resolved! Poe really is a master of the short gothic story.