This story is full of cops (six in all) but we don't meet the one in the title, the one who finally arrests Soapy, until the end of the story. Same thing with the anthem—we learn what this mysterious-sounding thing is near the ending. We can't really talk about the title without looking at this whole anthem situation, so here's what we've found out about that:
The Anthem
And the anthem that the organist played cemented Soapy to the iron fence, for he had known it well in the days when his life contained such things as mothers and roses and ambitions and friends and immaculate thoughts and collars. (42)
It's not clear if the organist is also singing the hymn or if Soapy just recognizes the tune of it. What's more important is that this is the first time we've been given information on his past. We know that he is homeless, has been so for several years, and that he's spent the past few winters in the jail on Blackwell's Island. What we don't know is how he got this way. The anthem triggers memories of his life when it was better.
We learn that Soapy used to have a nice life, one that included church, "immaculate thoughts," and collars, which are what priests and ministers wear. When he remembers this old life, Soapy is finally able to see his current life clearly, and he doesn't like what he sees:
He viewed with swift horror the pit into which he had tumbled, the degraded days, unworthy desires, dead hopes, wrecked faculties and base motives that made up his existence. (42)
This passage suggests that the choices Soapy has made have led to his current life. At the same time, it suggests that it happened by accident, that he "tumbled" into his situation. Maybe it was some combination of the two. Whatever the case, the anthem fills him with the power and the drive to turn his life around. He has no doubts, in this moment anyway, that he can do much better. We are told,
Those solemn but sweet organ notes had set up a revolution in him. To-morrow he would go into the roaring downtown district and find work. A fur importer had once offered him a place as driver. He would find him to-morrow and ask for the position. He would be somebody in the world. He would— (43)
The anthem helps Soapy want something more out of his life than what he has now. Has this ever happened to you? Has a piece of music or other art ever changed your life? Was the change lasting? Do you think Soapy's change will be permanent?
The Cop
The cop acts to thwart Soapy's newfound plans. Although Soapy has changed on the inside, to the sixth policeman, Soapy is still obviously a homeless man. Soapy isn't pulling anymore shenanigans at this point in the story, so he's probably arrested for "vagrancy"—the crime of not having a place to live and not being able to support himself.
For Soapy, getting what he used to want presents an obstacle to his new goals and plans. They cop might represent obstacles Soapy will have to overcome to reach his dreams (whatever those dreams are).
The Cop and the Anthem
We might look at the cop and the anthem as different parts of Soapy's life, or as two different paths available to him. The anthem represents the life he used to have, and wants to have again. The anthem represents a path where Soapy can live a good life on his own terms. Everybody has their own idea of what a good life is, but it usually doesn't include being in jail. Soapy's goal isn't really to go to jail, but to have a warm place to live for the winter. When he hears the anthem he feels empowered to achieve that goal without sacrificing his freedom.
The cop represents the life Soapy has now, one where he is either avoiding the police or looking for the police to arrest him, depending on the season. The cop represents a path where Soapy has to be under someone else's authority, where he has to do what others tell him to do.