Little bit of time before event (5)
I believe the answer is:
trace
'little bit' is the definition.
(a trace is a small amount)
'time before event' is the wordplay.
'time' becomes 't' (abbreviation).
'before' says to put letters next to each other.
'event' becomes 'race' (race is a kind of event).
't'+'race'='TRACE'
'of' acts as a link.
(Other definitions for trace that I've seen before include "Pinch" , "Make a copy using transparent paper" , "a piece of harness" , "Hunt down" , "Small quantity; find" .)