The fellow grew older but coped (7)
I believe the answer is:
managed
'coped' is the definition.
('manage' can be a synonym of 'cope')
'the fellow grew older' is the wordplay.
'the fellow' becomes 'man' (fellow can mean a man).
'grew older' becomes 'aged' ('age' can be a synonym of 'grow older').
'man'+'aged'='MANAGED'
'but' acts as a link.
(Other definitions for managed that I've seen before include "Controlled, was in charge of" , "Coped" , "Ran, administered" , "Supervised (staff)" , "Contrived" .)