What is the unique style of O'Henry's way of writing in "After Twenty Years"?
One might call this style the use of a twist or a surprise ending. O'Henry purposefully withholds the information that the policeman is Jimmy Wells, thus keeping Bob and the reader in the dark.
The reader and Bob only know what is presented in the text. Therefore, the reader and Bob find out the truth together at the end of the story. This style is the opposite of dramatic irony. Dramatic irony is when the reader knows more than the characters. This is not the case in this story. In this story, what we have is best described as situational irony: a situation where the outcome is unexpected: Bob and the reader are surprised by the outcome together.
The policeman (Jimmy) and the narrator know more than the reader. And although the narrator is omniscient, the narrator withholds information so we can say that this is an unreliable narrator (an unreliable narrator can be one who doesn't know everything going on or one who does but withholds information).
We are given a clue that the plain clothes policeman is not Jimmy Wells when Bob notes how much "Jimmy" has changed:
"You've changed lots, Jimmy. I never thought you were so tall by two or three inches."
The unreliable narrator withholds information which leads to a situation where the ending is unexpected, a twist, a kind of situational irony.