I feel that the narrator's behavior at the end of the story is selfish, uncaring, and despicable in a sense, as well. He is not being his "brother's keeper", especially since his brother is younger, weaker, sickly, and dependent on the older brother. The younger boy, called Doodle by the older brother, tries his best to keep up with the running pace of the older brother as they seek to get out of a rain storm. However, he is no match for his older brother's strength, resilience, and vigor - he tries valiantly to keep pace, but cannot.
Eventually, Doodle succumbs to the physical demands on his body and begins to bleed from his mouth, as if his heart has exploded within him. Doodle was born weak and sickly and the family didn't think he would live long in life and he had to be constantly watched for any threats to his existence because of his health problems.
I believe the narrator is at least partly responsible for his younger brother's death because he pushed him so hard to keep up with him by running so fast ahead of Doodle that the weaker boy had to fight with all his strength to try and catch up to his brother. Doodle may have died regardless due to his frail condition and because of the health problems he faced all of his life. Nevertheless, it seems that the older brother hastened the arrival of his death somewhat, so yes he bears some responsibility in this short story.
I believe the emotions that the older brother is experiencing at the end of the story are sorrow, guilt, as well as self-abhorrence for his actions. He feels sorrow for the actual death of his younger, weaker sibling, whom he realizes now is a human being worthy of love and care. He feels guilt because of the cruel, selfish, unthinking and uncaring way he dealt with his brother. He has self-abhorrence because he now sees his true character in a light he never really saw it in before, and he now hates what he has done, and probably hates himself for his terrible actions that contributed to this tragedy.