The narrator never gets a name, but we learn a fair amount about him in the opening paragraphs, and we follow the story solely from his perspective. At times, he appears more as a peripheral figure than a central character, since he acts mostly as a passenger while the old men on the ship make all of the active decisions (which he can't understand anyways).
That's no mistake. This story is all about fear, and experiencing these events through the filter of our narrator is a surefire way to ensure that we, too, feel his fear The first-person narration not only lets us see what he sees as he sees it (and in doing so makes the freak-out more immediate), but lets us spot flaws in the narrator's own opinion that set him up for all kinds of terror.
He starts out by establishing how rational and observant he is, with high falutin' lines like, "Upon the whole, no person could be less liable than myself to be led away from the severe precincts of truth by the ignes fatui of superstition" (1). Insert eye roll here. Then the sheer craziness of what goes down then pulls the rug out from under him. Rational? What kind of rational person sees ghosty sailors on a ship made of rotten wood?
Still, he clings to his sanity. Unfortunately, the narrator's rationality can't help him do anything more than record what happens in horrific detail, as well as reminding us that he could be killed at any moment. A third-person narrator would pull us away from that, and rob the tale of its in-your-face scares.