Student is finished, caught by rising impudence (5)
I believe the answer is:
pupil
'student' is the definition.
(synonyms)
'finished caught by rising impudence' is the wordplay.
'finished' becomes 'up' (similar in meaning).
'caught by' is an insertion indicator.
'rising' says the letters should be written in reverse (going up in a down clue).
'impudence' becomes 'lip' (lip is a kind of impudence).
'lip' backwards is 'pil'.
'up' put into 'pil' is 'PUPIL'.
'is' acts as a link.
(Other definitions for pupil that I've seen before include "Dark centre of the eye" , "Eye hole" , "One being taught" , "A student at school or eye part" , "feature in "Observer"?" .)