Am older, bursting to succeed to this? (7)
I believe the answer is:
earldom
'succeed to this?' is the definition.
'am older bursting' is the wordplay.
'bursting' is an anagram indicator.
'am'+'older'='amolder'
'amolder' is an anagram of 'EARLDOM'.
'to' is the link.
(Other definitions for earldom that I've seen before include "Peer's domain" , "Noble's status" , "Noble's lands or rank" , "Noble rank" , "Essex?" .)