Outlaw about to shout down primate (6)
I believe the answer is:
baboon
'primate' is the definition.
(I know that baboon is a type of Old World monkey)
'outlaw about to shout down' is the wordplay.
'outlaw' becomes 'ban' (banning is a kind of outlawing).
'about' is an insertion indicator.
'to shout down' becomes 'boo' (I can't justify this - if you can you should give a lot more credence to this answer).
'ban' going around 'boo' is 'BABOON'.
Can you help me to learn more?
(Other definitions for baboon that I've seen before include "Large long-nosed Arabian or African monkey" , "animal" , "Large primate" , "A large one of 17 across" , "Large long-nosed African or Asian monkey" .)