Actually, there is no identified narrator in this story, so it would not be possible to describe one. A story with an identified narrator is written in the first-person point of view; a narrator tells the story as he or she observed or experienced it from a personal point of view. Sometimes the narrator has a name, sometimes not, but the narrator will have an identity, and his or her identity will influence how the story was experienced and then related.
"An Occurrence at Owl Creek Bridge" is written in the third-person point of view. The voice telling the story is not a character with an identity; it is the creative voice of the author telling the story. Sometimes a third-person point of view is limited; the author will limit the details of the story to include only what an observer at the scene would see or hear. Sometimes a third-person point of view is omniscient; the author acts as "all-seeing," and includes not only what could be seen and heard, but also what one or more of the characters is thinking and feeling.
In "An Occurrence at Owl Creek Bridge," Bierce writes from both the limited and the omniscient third-person point of view, moving from one to the other in different parts of the story. For instance, most of part III of the story is written in the third-person omniscient point of view as Bierce takes the reader into Peyton's mind, but the conclusion of the story returns to the third-person limited point of view.