"Shooter" is actually perfectly correct. As a native speaker, I'm familiar with the use of the word "shot" to mean "shooter," specially a good shooter. For example, we can say either that Annie Oakley was a "sharpshooter" or that she was "a sharp shot." I don't think there's any logical explanation for the usage other than to say "this is just one of the meanings for the word 'shot.'" I double-checked a dictionary, and it says that one of the meanings of "shot" is "One who shoots in a particular way: 'a good shot with the rifle and the bow.'"

Hi! Well...the word shot in the sentence is a 'countable noun'. The expression 'he was a good shot' does mean that he 'is good at shooting', but instead it says, effectively, that each shot he takes is good. The sentence doesn't make perfect sentence when you break it down. My advise is to just learn the expression and not think too much about it in terms of grammar!