Poe does use an adjective/noun phrase to denote the "old man." But for the narrator, he only uses the pronoun "I." In doing so, the reader sees the events of the story through the narrator's crazy perspective. There is one other presence in the story, not really a character but an "auditor" - the person (and/or reader) to whom the narrator is speaking. The narrator is therefore confessing to this "you" which could be an auditor (some unknown listener), the reader, or even the narrator himself. In this way, the narrator is confessing to the reader (and/or himself) but, again, sticking to the "I" perspective, the reader gets into the madness of the narrator's mind. The narrator, "I," keeps trying to justify and explain to "you" the method and reason to his madness. So, this is a confession but also a defense.
And now have I not told you that what you mistake for madness is but over-acuteness of the senses?
Also note that the narrator is drawn to the old man as much as he despises the old man's eye. He goes to the old man's room night after night; the old man's eyes are always closed until the final night. The narrator, "I," is driven to and repulsed by the old man's eye; the homophones "I" and "eye" audibly show this relation between the narrator and the old man. This verbal, audible similarity is fitting because the narrator's other distorted sense (which the narrator says is acute rather than a symptom of his madness) is his sense of hearing. He is plagued by the old man's eye, a madness of seeing and a revulsion of being seen by the old man. When the old man is dead and hidden, the narrator is plagued by the alleged beat of the old man's heart. From sight to sound, the narrator is disturbed. Throughout the story, the narrator (I) tries to justify and/or confess his actions to a "you." Sticking just with "I," Poe gives the reader no background on the narrator; all we have is his madness and this is necessarily the one thing we can focus on. In other words, with no name and no background, readers are essentially forced to focus on mad perspective of the "I" (narrator).