The use of third-person narration in "After Twenty Years" by O. Henry affects the story as is described in the fourth option: it allows the reader to get to know each character.
Had it been the a first-person narration, options 2 and 3 could have been right. Narrating from a first-person perspective is a technique employed to create a bond between the narrating character and the reader, as well as to focus on that character's emotions and thoughts.
The first option is incorrect because this short story does not offer too much information to the point of confusing readers. Quite the contrary, it gives only enough information to prevent the reader from completely guessing what the ending will be.
Option 4, however, is the typical advantage of a third-person narration. It gives the writer the possibility of describing each character from a less personal and biased point of view. In this specific short story, readers get a brief physical description of both important characters (the cop and the man from the West). Not much is said on the characters' personalities or morals: that is left for the readers to conclude after the dialogs and the climax.
