Almost doesn't start in good time (5)
I believe the answer is:
early
'in good time' is the definition.
(being early is arriving in good time)
'almost doesn't start' is the wordplay.
'almost' becomes 'nearly' ('nearly' can be a synonym of 'almost').
'doesn't start' means to remove the first letter.
'nearly' with its first letter removed is 'EARLY'.
(Other definitions for early that I've seen before include "New-season" , "Infancy, when Wordsworth had 5 across of 5 down" , "with time to spare?" , "Before the usual time" , "Premature" .)