Angry boast of a self-important type (5)
I believe the answer is:
irate
'angry' is the definition.
(irate can mean angry or annoyed)
I cannot understand how the remainder of the clue works.
Can you help me to learn more?
(Other definitions for irate that I've seen before include "Cross, irritated" , "Annoyed, irritated" , "A tier (anag)" , "ranting" , "Livid" .)