Slowly run out of unopened alcohol (4)
I believe the answer is:
ooze
'slowly run' is the definition.
'ooze' can be an answer for 'run' (oozing is a kind of running). I'm not certain of the 'slowly' bit.
'unopened alcohol' is the wordplay.
'unopened' means to remove the first letter.
'alcohol' becomes 'booze' (synonyms).
'booze' with its initial letter removed is 'OOZE'.
'out of' is the link.
Can you help me to learn more?
(Other definitions for ooze that I've seen before include "Mud at bottom of river" , "Emit" , "Leak - slimy mud" , "... sweat" , "River-bed mud; leak slowly" .)