Cooler to be employed in backing Ono, Sir Paul (6)
I believe the answer is:
prison
'cooler' is the definition.
(cooler can informally mean a prison)
'employed in backing ono sir paul' is the wordplay.
'backing' is a reversal indicator.
'employed in' says the answer is hidden in the clue.
'nosirp' is hidden within 'ono sir paul'.
'nosirp' in reverse letter order is 'PRISON'.
'to be' is the link.
(Other definitions for prison that I've seen before include "Jail" , "Stir - can" , "cage" , "No rips (anag)" , "Place of detention for criminals" .)