The resolution to a story is typically defined as the part that comes after the climax of the story.  In this story, I think that the climax is the part where the sniper shoots and kills the enemy sniper.  We have followed the sniper's actions, wondering if he will live or die, now we find out.

The resolution of the story goes from there.  In it, we see the sniper's attitudes towards the war wavering -- he feels sick about what he's done and he hates the war.  Then he almost shoots himself accidentally and that snaps him out of that.  Finally, he finds out that he has killed his own brother (and we assume that will change his attitudes again).  So, in the resolution, we come down from the climax and we see the author lay out the message he is trying to impart in the story.

"The Sniper" does not have a true resolution because typically the resolution includes the tying up of loose ends at the end of a work.  In "The Sniper"--and in many other surprise ending stories--a formal resolution would ruin the effect of the plot's suspense.  Some of the loose ends that are left unresolved by O'Flaherty in his short story are the fate of the protagonist, his and his family's reaction to his brother's death, and most importantly, whether he tells his family that he was the one who killed his brother.

While many readers and viewers do not like works without closure, sometimes closure (or resolution) must be forfeited in order to have a truly surprising ending or in order to force the reader/audience to formulate their own resolution.