How is the central conflict in "The Man Who Was Almost a Man" resolved?
This all depends on what you think the central conflict is. I'll run through a couple different options for you.
If you feel that central conflict is Dave's accidental shooting of the mule, that doesn't really resolve itself. Dave agrees to work hard to pay off the debt for the mule, but then never follows through. So, he never repays or takes responsibility for the actions that he committed. The characters try to resolve the situation, but Dave's immature attitude thwarts their best attempts.
If you feel that the central conflict revolved around Dave wanting to earn respect and "manhood," that issue is also not resolved. He doesn't earn respect or manhood. Instead, he lies, cheats, shirks responsibilities, and essentially proves how immature and untrustworthy he is. He never feels that respect and honor that a man would; in fact, he skips town to escape having to feel like a little kid anymore. At the beginning he tries to resolve his inner conflict through attaining the gun; it just doesn't really work out for him.
This story falls under the style of modernism, which is a type of writing that often had storylines where conflict wasn't resolved. Modernist writers liked experimenting with traditional forms of storytelling; traditional forms usually resolve conflicts and have happy endings. This one doesn't. The conflicts remain, and Dave leaves, leaving the reader with a sense of unfulfillment and frustration. I hope that those thoughts helped a bit; good luck!