First appearance of rime, peer curiously (8)
I believe the answer is:
premiere
'first appearance' is the definition.
(eg first performance of a play)
'rime peer curiously' is the wordplay.
'curiously' indicates anagramming the letters.
'rime'+'peer'='rimepeer'
'rimepeer' is an anagram of 'PREMIERE'.
'of' is the link.
(Other definitions for premiere that I've seen before include "Leading lady" , "Mere pier (anag.)" , "significant performance" , "First screening" , "Opening performance" .)