The Shooting range in Westcliff will appear in your quest log and stay there until you manage to snag the top prize. It is not an easy thing to do by any means and will require a lot of practice or a ludicrous amount of luck.
There are no real strategies to to beat the Shooting Gallery as it pretty much requires you to be awesome at shooting (which I am not!). One thing that made things a little easier in my attempts was to use a clockwork pistol as it has a fast rate of fire and reload time. Addiitonally, you want to ideally aim for the heads of your targets for bonus points.
Again, it will take a bit of practice (or a hell of a lot in my case!) but is not impossible. The prizes for the shooting range depend on the score you achieve and the top prizes is a Legendary Weapon - The Red Dragon (Pistol). Prizes are distributed as such:
|65||Worn Double Bed (Furniture)|
|150||Hero Doll (Used for an achievement)|
|175||Legendary Weapon - The Red Dragon (Pistol)|
Note that it is possible to earn the top prize a second time, but in place of the legendary weapon, the game will reward you with 5,000 gold.