Celebrity set off, last out (4)
I believe the answer is:
star
'celebrity' is the definition.
(I know that star is a type of celebrity)
'set off last out' is the wordplay.
'set off' becomes 'start' (synonyms).
'last out' means to remove the last letter (I've seen 'last out of' mean this).
'start' with its final letter taken away is 'STAR'.
(Other definitions for star that I've seen before include "Famous entertainer" , "A heavenly body or film favourite" , "One guiding" , "Eg, Betelgeuse" , "Famous player" .)