The point of view in this story is the third person point of view. Simple clues to that point of view are the pronouns that are used in the story. Words like "he" and "she" are often good indicators of third person narration. The story is about a sniper, and the narration continually refers to the sniper using "he" and "him."
He was eating a sandwich hungrily. He had eaten nothing since morning. He had been too excited to eat. He finished the sandwich, and, taking a flask of whiskey from his pocket, he took a short drought.
If the story was told in first person, readers would be reading the narration from the sniper's perspective, and we would be reading sentences that use "I" to refer to the sniper.
Despite being a third person narration, the story is very much limited to the sniper's perspective. This is called a third person limited point of view. An omniscient point of view would give readers the thoughts of every character in the story, but that doesn't happen. We never know what is going on inside the mind of the old woman, the enemy soldier, or the Free Stater sniper. The limited narrative point of view helps tie readers to the emotions of the Republican sniper. It also lets readers view the enemies as exactly that. They are faceless and nameless enemies. They are targets to be shot and killed. An omniscient narration would have ruined that because readers might be saddened when the "enemy" sniper is killed.
Liam O'Flaherty's short story, "The Sniper," is written using a third person limited narrator. A third person limited narrator is very different from a third person omniscient narrator. A third person omniscient narrator knows everything about every character depicted in the story. On the other hand, a third person limited narrator only knows everything about one character (typically the protagonist). A third person narrator does not have a part in the text either; he or she is only relaying the story for the reader.
The importance of the third person limited narrator in O'Flaherty's story lies in the ending. Over the course of the text, readers come to know the sniper relatively well. They come to identify him as disciplined, hardened by war, and very good at his job. If a third person omniscient narrator would have been used, readers would have known (dramatic irony) about the identity of the other sniper (the protagonist's brother). This would have ruined the surprise for both the reader and the sniper himself. Given the emotional detachment of the narrator, the shock of the ending proves surprising.