Write up part of Bible first for primate (5)
I believe the answer is:
potto
'for primate' is the definition.
(potto is a kind of primate)
'write up part of bible first' is the wordplay.
'write up' is a reversal indicator.
'part of bible' becomes 'ot' (Old Testament).
'first' becomes 'top' (top can mean the best or first).
'ot'+'top'='ottop'
'ottop' reversed gives 'POTTO'.
(Other definitions for potto that I've seen before include "Kinkajou" , "Little creature" , "Slow-moving African" , "Nocturnal primate" , "Sort of 13 [LEMUR]" .)