Public school type’s rising importance in surrounding area (7)
I believe the answer is:
etonian
'public school type's' is the definition.
'rising importance in surrounding area' is the wordplay.
'rising' is a reversal indicator (going up in a down clue).
'importance' becomes 'note'.
'surrounding' means one lot of letters goes inside another.
'area' becomes 'a' (maths abbreviation).
'note' backwards is 'eton'.
'in' enclosing 'a' is 'ian'.
'eton'+'ian'='ETONIAN'
(Other definitions for etonian that I've seen before include "Pupil at posh English school" , "One educated at famous public school" , "Public school boy" , "Relating to one particular school" , "Alumnus of famous English school" .)