The most common and most likely answer to this crossword clue is the 9 letter word POOTERISH
I've seen this crossword clue in Independent
P
O
O
T
E
R
I
S
H
More clues leading to the result POOTERISH
Buddy explains!
The straight part
We know that "Self-important" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case self-important = pooterish.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.