School bill's rising (4)
I believe the answer is:
eton
'school' is the definition.
(famous UK public school)
'bill's rising' is the wordplay.
'bill' becomes 'note' ('note' can be a synonym of 'bill').
'rising' is a reversal indicator (going up in a down clue).
'note' back-to-front is 'ETON'.
(Other definitions for eton that I've seen before include "British public school of note" , "Berks college" , "learn here?" , "Berkshire public school" , "English college of note" .)