Drunk - also strong and self-important (4,3,6)
I believe the answer is:
high and mighty
'self-important' is the definition.
(I've seen this before)
'drunk also strong' is the wordplay.
'drunk' becomes 'high' (I've seen this before).
'also' becomes 'and' (both can mean in addition).
'strong' becomes 'mighty' (I am not sure about this - if you are sure you should believe this answer much more).
'high'+'and'+'mighty'='HIGH-AND-MIGHTY'
'and' is the link.
Can you help me to learn more?
(Other definitions for high and mighty that I've seen before include "Acting as if one believes one is very important" , "Stuck up" , "Important, influential and arrogant" , "like the Sun, perhaps? - with 9a,10a" , "Domineering" .)