This mission has you coming for a duel with the Sheriff of Lynchwood. Aside from being filled with cliches like how they are going to use pistols, that you're supposed to come alone but the bad guy comes with goons all over the rooftops and that one of the parties is the sheriff, there's also an interesting reference to the song "I Shot the Sheriff".
After you make it to the place where you are going to have the duel, you'll be given two optional objectives. The second of the two reads "But do not shoot the deputy". The chorus of the song goes "I shoot the sheriff/But I did not shoot the deputy." Since you could theoretically kill the sheriff with melee attacks, the first optional objective which requires you to use a pistol also requires you to shoot the sheriff meaning that if you satisfy both objectives, you will have shot the sheriff but not the deputy.
As for other connections, the Sheriff has every reason to hate but specifically says that she likes you. In the song it is quite the opposite. The Sheriff hated the person singing and had it in for him. In this regard, it is safe to say that both Sheriffs wanted the protagonist dead.
