The speaker has come to town with one aim: to blot out the hours. He wants drink to wipe away time, and with it whatever he would rather not feel. But the second stanza admits a letdown. The drinking doesn't seem as good as it did at first, which hints at a habit that is wearing thin, giving less relief each time.
The last two lines are the strangest and saddest. He calls the bottle a woman, and what he hopes for is not love or comfort but that she "may let me be." The wording turns the bottle from a friend into something that has a hold on him, and he wants to be left alone by the very thing he seeks out. The title's word "progress" is bitterly ironic, since nothing here moves forward. The mood is weary and quietly resigned. The short, plain lines and the lack of drama make the confession feel honest and a little hopeless.