Stiffly formal first person’s behind pair (4)
I believe the answer is:
prim
'stiffly formal' is the definition.
(prim can mean proper or formal)
'person's behind pair' is the wordplay.
'person' becomes 'im' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'behind' says to put letters next to each other.
'pair' becomes 'pr' (abbreviation).
'im' put after 'pr' is 'PRIM'.
'first' acts as a link.
Can you help me to learn more?
(Other definitions for prim that I've seen before include "Strait-laced - prudish" , "Old-maidish" , "Prudish, formal" , "Stiffly correct" , "Proper (type of rose?)" .)