'Shot in the neck' is a phrase used in both England and America to mean drunk since at least the late 1800s. While it is now archaic, it may not be as archaic as commonly thought, as it is commonly used in Regency historical novels, and these are by far the most common use of the idiom in written works.
The word 'shot' to refer to an alcoholic drink has been with us since at least the 1670s (although the specific meaning of a small dose of strong alcohol does not appear until the 1920s), building on the sense of shot meaning to 'throw something down'. There were a slew of idioms of this sort, including 'shot in the head', 'shot in the mouth', 'shot to ribbons', all meaning properly drunk.
In 1922 the competing idiom 'a shot in the arm' appears, refering to an injection of a stimulant -- essentially opposite in meaning. This may be the source of a rare, and apparently new English idiom, "not shot in the head". To be 'not shot in the head' about something is to be not very excited about it.
Regardless, the overwhelming usage of this phrase in recent years has been to talk about people who have indeed been literally shot in the neck; this is probably one of the biggest factors in its decline in use as an idiom; for an idiom to work, the literal meaning has to be uncommon enough that you don't throw people into a panic when you use it.