Quick to fly into a rage after closing time (5)
I believe the answer is:
eager
'quick' is the definition.
(thesaurus)
'into a rage after closing time' is the wordplay.
'into' is an insertion indicator.
'closing' indicates one should take the final letters (the letter which ends or closes the word).
'time' becomes 'age' (an age is a period of time).
The final letters of 'rage after' are 'er'.
'er' going around 'age' is 'EAGER'.
'to fly' is the link.
This may not be right. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for eager that I've seen before include "Anxious" , "Impatient" , "Swiss mountain" , "willing" , "Longing to see" .)