Dispiriting experience, but not at first, for possessor (5)
I believe the answer is:
owner
'for possessor' is the definition.
(synonyms)
'dispiriting experience but not at first' is the wordplay.
'dispiriting experience' becomes 'downer' (a depressing situation or person).
'but not at first' means to remove the first letter.
'downer' with its first letter removed is 'OWNER'.
(Other definitions for owner that I've seen before include "One who has something" , "Man of property" , "Title-holder" , "Holder, possessor" , "Person having property" .)