In "After Twenty Years," what kind of area was the policeman patrolling?

The area seems to be made up of two-story wooden buildings with shops on the ground floor and offices on the second floor. Property was not as valuable in New York in O. Henry's time. Nowadays the same neighborhood would be covered with high-rise buildings. New York was constantly undergoing an evolution in its architecture. Twenty years before the story opens the neighborhood must have been more lively. There were probably plenty of saloons as well as restaurants like "Big Joe" Brady's. Now it is a neighborhood where people in the office go home at five or six o'clock and the stores lock up at around the same time. 

O. Henry wanted the encounter between Jimmy Wells and "Silky" Bob to take place in a dark, nearly deserted section of the city. It had to be dark so that Bob would not recognize the policeman as his old friend. It also had to be very quiet at that time of night, about ten o'clock, so that Bob might seem a little suspicious standing in a darkened doorway. The two men wouldn't have made an appointment to meet in front of a hardware store. They had made the appointment to meet at the same place where they separated, at "Big Joe" Brady's restaurant. O. Henry had to explain that Bob had found that the restaurant had been torn down and replaced by some shops.

Bob thinks the uniformed cop is approaching him because he looks suspicious standing in the shelter of a darkened entranceway to a closed hardware store. He is mistaken. The cop is there to keep the appointment he made to meet Bob on that spot in twenty years. Jimmy doesn't get a chance to identify himself. Bob is a big talker and starts explaining his presence there before Jimmy can say anything.

"It's all right, officer,” he said, reassuringly. “I'm just waiting for a friend. It's an appointment made twenty years ago. Sounds a little funny to you, doesn't it? Well, I'll explain if you'd like to make certain it's all straight. About that long ago there used to be a restaurant where this store stands— ‘Big Joe’ Brady's restaurant.”

Then, before Jimmy can get a word in edgewise:

The man in the doorway struck a match and lit his cigar. The light showed a pale, square-jawed face with keen eyes, and a little white scar near his right eyebrow. His scarf pin was a large diamond, oddly set.

We learn later that as soon as Bob lights his cigar, Jimmy recognizes him as the man wanted by the Chicago police. After that, Jimmy decides not to introduce himself. Jimmy just listens to what Bob has to say, which conveys a lot of background information to the reader in dialogue form. Jimmy decides that he can't arrest Bob himself but that he will go back to the station house to get someone else to do it. He has to make sure Bob will be waiting there. 

“I'll be on my way. Hope your friend comes around all right. Going to call time on him sharp?”

“I should say not!” said the other. “I'll give him half an hour at least. If Jimmy is alive on earth he'll be here by that time. So long, officer.”

Because the neighborhood is dark, it is easy for the plainclothes officer to pose as Bob's old friend for at least a short time. The wet weather explains why the officer has his overcoat collar turned up to his ears, hiding part of his face. It also explains why Bob is standing deep inside the entranceway to the hardware store. He wants to smoke his cigar and can't do that in the rain. And he has to remain there until he finishes the cigar. The neighborhood is created to suit the plot, and so is the bad weather.