The clue's in the title: "Why I Live at the P.O." (emphasis added). The story's a monologue, which, like all monologues, is told in first-person narrative. Essentially, the narrator is telling us the long, convoluted tale of how she came to live in the post office where she is the postmistress. The first-person narrative is arguably more effective for the telling of what is basically a shaggy-dog story. It's like we're listening to a stranger telling us their life story on a park bench or in a bar. The events that Sister relates are so absurd, so ridiculous and hard to believe, that any credibility they may have can only come out through a first-person narrative. Any other perspective would simply convey the fundamental absurdity of Sister's tall tale without in any way forcing us to ask ourselves whether anything she says is actually true.

The short story is narrated by an unnamed first person narrator in the first person.

First person point of view is when a narrator describes things from his or her own perspective, using first person pronouns such as I, my, me, and we. 

First person is closer, but less reliable.  We see this in this story, as we hear Sister’s perspective and not Stella-Rondo’s.

But I said she couldn't convince me though she talked till she was blue in the face. 

Yet as we hear the narrator’s version of the story, where she constantly thinks other people are in the wrong, we get the idea that she might be the one who’s wrong.  In this case, the first person narration actually adds to the drama of the story, because we do not trust the narrator.