Doc is certainly much more experienced than I am, but do yourself a favor an believe me in this case. Don't try to go with a plain LDR. Use a real IR detector, such as the one that Grumby_Mike suggested. If you go with a LDR, you will need a whole lot of additional electronics (noise filtering, amplifier etc) just to distinguish your signal from the ambient IR noise.
I actually built a couple of month ago something almost exactly like what you are going to build. I newer quite finished it, but I got the gun and the target working quite well. The shooting distance was about 10-15 feet. From that distance the IR beam diameter on the target was about 2 inches. (Depends on the length and the diameter of the gun's barrel.)
At first I experiment with a photo transistor. It kind of worked at night time, but at day time when sunlight came through windows, it didn't work at all. With an IR detector sunlight wasn't a problem.
