The Republican sniper stationed on a rooftop near the O’Connell Bridge witnesses an old woman wearing a tattered shawl approach and speak to a Free Stater soldier manning a turret attached to a vehicle. He then sees the old woman point towards his position on the rooftop and realizes that she is an informant. The old woman has just given away his position to the Free Stater soldier and he is now in danger of being shot. The Republican sniper reacts by shooting and killing both the man in the turret and old woman running across the street. Despite successfully killing the enemy informant and soldier in the turret, the Republican sniper attracts the attention of an enemy sniper stationed on the opposite rooftop. Immediately after shooting the soldier and old woman, the Republican sniper is shot in the forearm and is forced to improvise in order to survive and kill the Free Stater sniper.

The sniper kills the old woman because she is an informant for the Free Staters. 

In addition to her being an informant, the old woman knows the location of the protagonist sniper.  The sniper watches her flag down an enemy vehicle.  She talks to the soldier inside, and she points toward where the sniper is.  At this point in the story, three people know the location of the Republican sniper.  The enemy sniper, the old woman, and the soldier in the vehicle all know where he is hiding.  In order to safely exfiltrate himself from his location, the sniper needs to kill everybody that knows where he is.  The sniper successfully shoots and kills the old woman and the soldier on the street.  Unfortunately, taking those shots exposed him to the enemy sniper, and the protagonist is wounded.  The wound prevents him from being able to effectively hold his rifle.  In order to survive, the protagonist comes up with a plan that allows him to use his sidearm to kill the enemy sniper.   

why does the sniper kill the old woman and what happens to him after he kills her ?

The sniper, from The Sniper by Liam O'FLaherty, kills the old woman in the street because she is pointing out his position to the other side.  She is an informer.  Because she is putting his life in danger, he shoots and kills her.  This is war, and even cilvilians, like the old lady, are fair game.

Immediately after he kills the old woman, another sniper on a rooftop across the street knows his position and fires.  The sniper is hit.  He is shot in the arm, which renders his arm useless.