It would be difficult to determine the socio-economic class of the sniper himself since this story takes place during the Irish Civil War and soldiers came from many economic classes. He is a Republican, which means he was fighting for the government, but there are not enough hints in the story to tell the reader what the sniper's socio-economic class was.
However, it is possible to give the socio-economic background of the place in which they are fighting. First of all, it is in the city of Dublin, the capital of Ireland and seat of its government. At the end of the story, the sniper "peers around the corner into O'Connell Street". O'Connell street is the main street of Dublin. In the first paragraph it says that "Around the beleaguered Four Courts the heavy guns roared." The Four Courts building housed the four courts of Ireland: the Chancery (Circuit Court, handling licensing,taxation, and any of the King's affairs), Exchequer (a superior court that dealt mostly with revenue), King's Bench (the Supreme Court), and Common Pleas ( subject vs. subject). The building was designed by James Gandon, who was the architect of the Customs House, although it was heavily damaged in this civil war. At one time this building was considered one of the finest buildings in Dublin.
Then at the beginning of the second paragraph it says "On the rooftop near O'Connell Bridge, a Republican sniper was watching." The O'Connell Bridge was also designed by James Gandon. It flows over the River Liffey, which is also named in paragraph one.
Based on the buildings and streets surrounding this story, it appears to be an upper class neighborhood. The buildings have designers, government officials did business here, and it is the center of the capital. You will note that it says "Everywhere around was quiet. There was not much danger in going through the streets". All the important people had fled to safety, and the ones left were the soldiers.