The climax of a story is considered to be the moment of the most tension, the moment when the main conflict is at its height, and it can even be a turning point of some sort. It is then helpful to determine what the main conflict is before trying to ascertain the climax of a story. As much of the story revolves around the conflict between the Republican sniper and the enemy sniper on the opposing rooftop, it can be argued that the climax of the story occurs when the protagonist finally defeats his enemy, killing him. However, the sniper seems to experience a conflict with all three individuals he kills: the driver of the armored car, the woman informant, and the enemy sniper across the street. These conflicts do not arise out of a personal dislike for these individuals; rather, they arise out of the fact that he is participating in the Irish Civil War and must kill or be killed. After he kills the enemy sniper, the "lust of battle died in him. He became bitten by remorse." He feels revolted by the sight of his enemy's shattered body and tosses his own gun to the ground. This would seem to indicate that his main conflict is actually with society, a society which is so deeply embroiled in war.
It is for this reason that the protagonist's killing of the enemy sniper is not the story's climax, as the conflict isn't really between the two snipers but, rather, with the protagonist's conflict with society as a whole. Thus, the climax of the story is the final line: when the sniper turns over his enemy's body and "look[s] into his brother's face." It is in this moment that both the reader and, we can assume, the main character understand the true nature of the story's conflict, when it comes most intensely forward. The sniper doesn't hate his brother, but he's been compelled to hate his enemy by his society, and now he's killed his own family member as a result.