"Eveline" is an early Joyce story, and though he has not yet developed full stream-of-consciousness narration, he is interested in and centers on character thoughts. During this period of fiction writing, narration shifted from realistic narration centering on what happens in characters' lives, to omniscient narration centering on how characters experience what happens to them. Fiction centers on character thoughts in order to reveal how humans experience existence.
Thus, "Eveline" is third-person omniscient. The story centers on her thoughts. The reader reads of her inner conflict, her attitudes, her memories. She's conflicted about her role as caregiver--a role usually reserved for a mother, not a sister--and her desire for a better life. She thinks of the neighbor's house--he's an invading Protestant from the North. She remembers her mother.
In the end, the reader sees "Eveline" freeze, literally and figuratively, as she fails to escape and chooses to remain paralyzed and blinded in Dublin.