Graduate has rum cocktail in old country (5)
I believe the answer is:
burma
'old country' is the definition.
(I've seen this before)
'graduate has rum cocktail' is the wordplay.
'graduate' becomes 'BA' (someone with a Bachelor of Arts degree).
'has' means one lot of letters goes inside another.
'cocktail' indicates an anagram.
'rum' anagrammed gives 'urm'.
'ba' going around 'urm' is 'BURMA'.
'in' is the link.
(Other definitions for burma that I've seen before include "Rumba (anag)" , "Alternative name for Myanmar" , "Country no longer" , "Country also known as Myanmar" , "Another name for Myanmar" .)