I figured out a clever way to shoot a bow and arrow once. (Keep in mind, though, I wouldn't suggest using a regular stick for an arrow. That's all I had at the moment).

Also what Galactic suggested, cutting the arrow out of one bow, and cutting the bow out of another. Then you have an empty bow and a single arrow, which you can animate more realistically.

There are numerous ways to animate an arrow in LEGO.

1. You could mask the arrow, which is by far the easiest and most effective way of animating. If you want to make it fancier, you could add motion blur to the arrow, but that would require a full mask of the object which takes a good amount of time.

2. You could use string to move the arrow. However, you would need to light the set perfectly as so not to show the string. You also need to keep the arrow constant throughout the shot, which is difficult to do with string.

3. You could use CGI to animate the arrow. This is the most difficult option, due to the learning curve, time of producing the arrow, and realism of the composite on the final film.

It doesn't really matter which method you use, but just know that some methods are more difficult/longer than others.