This is a very interesting question and the answer of course depends on how you determine the word "hero". My initial impression is to say that we are not given enough information about the man to indicate if he is heroic or not - we know little of his reason for fighting, for instance, or his character, apart from his skill in battle.
Yet if you consider a hero to mean that someone is able to outwit his opponent in battle, this would indicate that the main character of this short story is a hero - he is able to cunningly entice the other sniper to reveal himself by feigning death:
Crawling quickly to the left, he peered up at the corner of the roof. His ruse had succeeded. The other sniper, seeing the cap and rifle fall, thought that he had killed his man. He was now standing before a row of chimney pots, looking across, with his head clearly silhouetted against the western sky.
Of course, being able to coolly think of a plan to outwit your enemy whilst wounded and then to carry it out successfully could be considered one aspect of being a hero.
However, on the other side, you will want to consider the man's actions in cold-bloodedly killing the informer, who is described as "an old woman, her head covered by a tattered shawl." Killing an unarmed civilian suddenly by surprise is definitely not an heroic action. Also important to note is what the sniper thinks of himself once he knows he has killed the other sniper:
The sniper looked at his enemy falling and he shuddered. The lust of battle died in him. He became bitten by remorse. The sweat stood out in beads on his forehead. Weakened by his wound and the long summer day of fasting and watching on the roof, he revolted from the sight of the shattered mass of his dead enemy. His teeth chattered, he began to gibber to himself, cursing the war, cursing himself, cursing everybody.
This appears to be a critical moment in the narrative when the sniper has a moment of epiphany, or sudden insight, and realises the futility and the inhumanity of the war. Is he a hero? Well, he certainly does not think he is at this moment, which of course foreshadows the revelation that his enemy is his brother.