Western Showdown is the fifth event in the Wild West booth in free play. In order to select it in free play you'll need to clear the four events before it, or you can just play Adventure Mode which will cycle to the booth eventually without needing to unlock it.
Western Showdown is a shooting range game. On the right building is a clock. There are a total of four targets attached to this building, which will cycle one at a time as you hit them. Each time you complete a four target cycle the clock will move ahead five minutes. Since the clock starts at 20 til noon, you'll have to do the cycle four times.
The buffalo appear in the center of the screen, first as a single buffalo, and then as a group of two buffalo. What you need to do is not shoot the third buffalo until the clock hands are showing noon. To do this, first concentrate on killing the first two buffalo as they appear. Then, hit the target cycle on the clock building four times until the hands point at noon. Then, shoot the third buffalo.
The achievement should pop as soon as you shoot the buffalo.