Let's take a look at that section of the story to better understand how the Sniper responds to being shot. I'll choose a few important quotes to illustrate:
"His forearm was dead. "I'm hit," he muttered."
Notice how he doesn't get frantic or start screaming or crying or whatever. He calmly thinking to himself, "I'm hit." A very cool customer, this fella is. Pragmatic and practical.
"The blood was oozing through the sleeve of his coat. There was no pain--just a deadened sensation, as if the arm had been cut off."
Again, very practical description of events. Neither the sight of the blood nor his non-functioning arm seems to phase him too much. He appears (at this moment, at least) very much in control of his passions.
"Quickly he drew his knife from his pocket, opened it on the breastwork of the parapet, and ripped open the sleeve. There was a small hole where the bullet had entered. On the other side there was no hole. The bullet had lodged in the bone. It must have fractured it. He bent the arm below the wound. the arm bent back easily. He ground his teeth to overcome the pain."
He knows what to look for when wounded. He is either well trained or experienced in this type of wound. He inspects the bullet hole with a very business-like method and is able to think rationally about the injury. He is a logical thinker, for the most part.
"Then taking out his field dressing, he ripped open the packet with his knife. He broke the neck of the iodine bottle and let the bitter fluid drip into the wound. A paroxysm of pain swept through him. He placed the cotton wadding over the wound and wrapped the dressing over it. He tied the ends with his teeth."
Again, very much in control of the situation. Knows just what to do and how to do it. He is able to overcome a very difficult situation by relying on training and force of will.
"The sniper lay still for a long time nursing his wounded arm and planning escape. Morning must not find him wounded on the roof. The enemy on the opposite roof coverd his escape. He must kill that enemy and he could not use his rifle. He had only a revolver to do it. Then he thought of a plan."
This is the last quote that refers to his wound. He is again thinking ahead. He knows the danger he is in, and despite the pain his arm is causing him he does not quit. He begins to formulate a plan. He is tough, practical, and has a very strong force of will.
That should about answer your question, but I wanted to add one more thing to round out the picture. It doesn't happen directly after he's wounded, but it's important:
"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 helps you to see that "The Sniper" is not without feelings, he has just suppressed them to deal with the reality at hand. I think that's an important point to remember and it really gives the guy a lot more character than if he just acted like a total robot.