Very keen shooter – he's foremost in game (4-2)
I believe the answer is:
gung-ho
'very keen' is the definition.
'gung ho' can be an answer for 'keen' (I've seen this before). I'm not certain of the 'very' bit.
'shooter he's foremost in game' is the wordplay.
'shooter' becomes 'gun' (I've seen this before).
'foremost' indicates taking the first letters.
'in' means one lot of letters goes inside another.
'game' becomes 'go' (go is a board game).
The first letter of 'hes' is 'h'.
'h' placed inside 'go' is 'gho'.
'gun'+'gho'='GUNG-HO'
(Other definitions for gung-ho that I've seen before include "Excessively enthusiastic" , "Stupidly aggressive" , "Overly zealous" , "Recklessly enthusiastic" , "Full-on, maybe jingoistic" .)