Make progress in good public school (3,2)
I believe the answer is:
get on
'make progress' is the definition.
(I've seen this before)
'good public school' is the wordplay.
'good' becomes 'g' (abbreviation).
'public school' becomes 'eton' (I've seen this before).
'g'+'eton'='GET ON'
'in' acts as a link.
(Other definitions for get on that I've seen before include "Board vehicle or make progress" , "fare" , "Hit it off" , "Advance; prosper" , "Board - prosper" .)