At great cost, heading off in good time (5)
I believe the answer is:
early
'in good time' is the definition.
(being early is arriving in good time)
'at great cost heading off' is the wordplay.
'at great cost' becomes 'dearly' (synonyms).
'heading off' means to remove the first letter.
'dearly' with its initial letter taken off is 'EARLY'.
(Other definitions for early that I've seen before include "Opening" , "Advanced" , "In good time" , "Gothic style following 13" , "Untimely" .)