Graduate, not up, getting stick (5)
I believe the answer is:
baton
'stick' is the definition.
(term for a stick or staff)
'graduate not up getting' is the wordplay.
'graduate' becomes 'ba' (someone with a Bachelor of Arts degree).
'up' is an anagram indicator (letters get chucked up).
'getting' says to put letters next to each other.
'not' anagrammed gives 'ton'.
'ba'+'ton'='BATON'
(Other definitions for baton that I've seen before include "A conductor wields it - or a policemen" , "Short stick" , "Stick passed in relay races" , "Conductor's wand" , "Piece of equipment" .)