Apprentice possibly fit to move up (4)
I believe the answer is:
lift
'move up' is the definition.
(synonyms)
'apprentice possibly fit' is the wordplay.
'apprentice' becomes 'l' (abbreviation for 'Learner').
'possibly' indicates an anagram.
'fit' anagrammed gives 'ift'.
'l'+'ift'='LIFT'
'to' acts as a link.
(Other definitions for lift that I've seen before include "Raise - shot in the arm" , "Raise the vertical conveyor" , "Exalt" , "It's an elevator in USA" , "Pick up" .)