The conjunction of Soapy's receptive state of mind and the influences about the old church wrought a sudden and wonderful change in his soul.

Further, Soapy is developed through his thoughts, actions, and speech and the reactions of others (indirect characterization). For instance, Soapy has no great "hibernatorial ambitions," but he does hope to go jail "on the Island" for the winter months and remain warm. So, he walks up Broadway to a "glittering cafe," hoping to enter, order an expensive meal, and then be arrested for not being able to pay the bill, thereby being sent to the Island. But, he is rejected by the head waiter, who spots his ragged pants.

After his first idea fails, Soapy throws a cobblestone through the window of a shop. When the police asks where the man who has committed this crime has gone, Soapy candidly admits, 

"Don't you figure out that I might have had something to do with it?"

But the policeman refuses to believe Soapy (another character's reactions). So, he enters another restaurant and eats heartedly. After his meal, Soapy admits that he has no money, but instead of being arrested, he is manhandled and thrown into the street. Soapy's further revised plans then turn in his flirting with a woman in hopes that she will scream and cause him to be arrested. But, again Soapy fails because the woman is a street walker and thinks he wishes to have her services. So, he "danced, howled, raved, and otherwise disturbed the welkin." But, when the policeman on the beat arrives, he thinks that Soapy is "one of them Yale lads celebrating" the win over Hartford College.

Now, Soapy remains defeated as he  steals a man's umbrella; however, it is not really the man's umbrella, so the "ex-umbrella man retreated."  Disgusted that he cannot get himself arrested, Soapy mutters as he walks, and he throws the umbrella away.  However, when he hears an organist playing an anthem, Soapy recalls his earlier life and vows to "pull himself out of the mire." But, at this point, a policeman grabs him, asking "What are you doin' here?" When Soapy replies, "Nothin'," ironically, he is arrested and sentenced to the Island.

Soapy has a particular goal in life, and he's absolutely determined to achieve it. With the winter coming on, this vagabond needs to find a warm place to stay during the cold weather. He figures that the best place for this will be jail. The plan's pretty simple: all he has to do is commit a crime, get arrested, and head off to a nice warm jail cell for the winter. But Soapy's best laid plans come to nothing, as his increasingly desperate attempts to get arrested end in farcical failure.

Henry uses a very simple linear narrative in telling Soapy's story, and with good reason. Soapy's quest for a warm jail cell for the winter is kind of like a pilgrim's progress in reverse. He has a clear goal in mind and a very simple idea for achieving it. Henry could've chosen to tell the story in flashback, with a rueful Soapy reflecting on his experiences from prison. But that would've robbed Soapy's tale of its immediacy. The onset of winter injects an element of urgency into Soapy's quest. And that sense of urgency would've been lost if Henry had chosen to tell the story in any other way.

There is nothing particularly unusual about the structure of the story "The Cop and the Anthem."  Indeed, its structure is fairly straight-forward.  The tale is told in linear, chronological fashion; that is, events in the story happen in the order in which they actually happen to the character involved (no real flashbacks, just him remembering things on occasion.)  It's fairly standard...if you look, you'll see that the name "Soapy" is within the first few words of each paragraph.

That said, it is also slightly episodic in that each of Soapy's encounters is kind of neatly contained as its own little event in the story.  Little time is spent on the development of each of these into full-fledged elements and each exists only long enough to demonstrate how Soapy's goal of being arrested is thwarted.

It is a fairly strait-forward piece of writing, no less effective for it.