The old woman is an informer for the Free State troops.  Because she is an informer, she poses a threat to the sniper.  Early in the story, an armored vehicle shows up near the sniper's position.  It pulls up next to the old woman, and a soldier sticks his head out of the armored car in order to talk to the old woman.  The sniper sees the old woman point up toward his position, which means that she knows where he is.  

Then round the corner of a side street came an old woman, her head covered by a tattered shawl. She began to talk to the man in the turret of the car. She was pointing to the roof where the sniper lay. An informer.

The sniper, in order to protect his own life, shoots and kills the soldier in the armored car.  He quickly follows that shot up by killing the old woman.  Unfortunately, in order to shoot both targets, the sniper exposes himself enough to be shot in the arm by the enemy sniper.    

The old woman is most certainly a sympathizer of the Free State troops who are fighting Republican troops during the Irish Civil War. The war broke out when the Irish Republican Army splintered after the signing of the Anglo-Irish treaty which made Ireland a part of the British Commonwealth. Republicans opposed the treaty and were soon fighting men who had once been part of their army. The old woman is in the process of pointing out the Republican sniper's position to the armored car commander, who is probably not police, but rather a soldier from the Free State army. The sniper kills both the commander, who foolishly sticks his head out of the armored car, and the old woman. The sniper seems cold and calculating in mechanically killing the two people. Later, of course, we find out that he also kills his own brother, who had been fighting as a Free State sniper. 

In this story, the sniper shoots the old woman because she is an informer.  He shoots her because she is trying to get him killed.

At the point in the story where he shoots her, an armored car has just appeared, carrying police.  The old woman talks to one of the men in the car.  She points up to where the sniper is.  The man looks out, the sniper shoots him.  Then he shoots the woman.

Even though the woman is from his neighborhood, she is an enemy if she is going to tell the police where he is.