The rising action of a story typically begins after some inciting incident: an event that initiates the conflict. O'Flaherty provides exposition, or background information, when the narrator describes the setting—June in Dublin during the Irish Civil War—as well as the protagonist: a young Republican sniper stationed on a rooftop.
The inciting incident occurs after he decides to take a risk and light his cigarette; another sniper, an enemy Free Stater, fires at him and nearly hits him with a bullet. Therefore, the rising action begins when the sniper puts out his cigarette and crawls away from his original position, hoping to escape the alert gaze of his enemy. This continues as the enemy sniper shoots again, as the armored car comes down the street, and as the female informant points out the protagonist's location.
The protagonist shoots the soldier in the armored car as well as the little old lady, but, as he does, the enemy sniper shoots and wounds him. He must come up with a plan, and he does, tricking the other sniper into believing that he's been killed. In this way, he gets the other sniper to reveal himself, and the protagonist shoots and kills his enemy.
I would argue that the climax of the story is actually the last line, and so the rising action ends here, when the sniper goes to find out who his enemy was and discovers that the dead man is actually his own brother.