Element of total bill I'm up for resolving outside (9)
I believe the answer is:
palladium
'element' is the definition.
(palladium is a kind of element)
'total bill i'm up for resolving outside' is the wordplay.
'total' becomes 'all' (I've seen this in another clue).
'bill' becomes 'ad' (advertisement stuck on wall).
'for resolving' indicates anagramming the letters.
'outside' is an insertion indicator.
'im'+'up'='imup'
'imup' with letters rearranged gives 'pium'.
'all'+'ad'='allad'
'allad' put into 'pium' is 'PALLADIUM'.
'of' is the link.
(Other definitions for palladium that I've seen before include "Theatre in London; rare metal" , "London variety theatre - element - safeguard" , "guarantee of protection" , "Metal element" , "Rare element" .)