The exposition of the story is the part that lets us know what is happening in the world. It shows what the story is going to be about.
To me, the exposition in this story goes from the start of the story up through the point where the narrator hears that the one little girl's family is fabulously wealthy and she will have a huge dowry.
I think there are two possible climaxes. One is the place where Mastakovich starts trying to kiss the 11 year old girl. This is surely the most emotionally gripping part of the story.
The other is where Mastakovich goes and talks to the girls parents and they are clearly so happy with his interest. At this point, we know who will "win" in this story.
The complication comes between the exposition and the climax and the resolution comes after -- particularly in the part where the narrator sees Mastakovich marrying the girl.