I thin the idea of Josh Hartnett's character is an introduction to the world, nay the city, of Sin City. It represents many ideas. People not always being what they seem. People dressed up are perhaps the most dangerous, people offering salvation not coming through the way you wanted. Deception, absolution, appearance. Don't trust those dressed to kill and only trust those who'd rather sleep rough and live wasted lives than get caught up in the deception of the city.
Of course it could be that they thought the movie was running short so they put in an extra scene with "squinting mumbler" Josh Hartnett. I'd ratehr believe that his scene serves as an introduction to the movie without launching into any story straight away, but you never know with these Hollywood types