Old shipping magnate’s donkey rides? (7)
I believe the answer is:
onassis
'old shipping magnate's' is the definition.
(20th-century shipping magnate Aristotle Onassis)
'donkey rides?' is the wordplay.
I cannot quite understand how this works, but
'donkey' could be 'ass' (ass is a term for a donkey) and 'ass' is present in the answer.
This explanation may well be incorrect...
(Other definitions for onassis that I've seen before include "Greek shipping magnate, d.1975" , "big name in shipping" , "Greek businessman" , "Aristotle -, married Jackie Kennedy" , "He married J. Kennedy" .)