Bill turned up around old ship (4)
I believe the answer is:
boat
'ship' is the definition.
(thesaurus)
'bill turned up around old' is the wordplay.
'bill' becomes 'tab' (tab is a kind of bill**).
'turned up' says the letters should be written in reverse.
'around' means one lot of letters goes inside another.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'tab' back-to-front is 'bat'.
'bat' enclosing 'o' is 'BOAT'.
(Other definitions for boat that I've seen before include "Water transport" , "Vehicle for travelling over water" , "Small ship" , "Vessel for travelling on water" , "Water craft" .)