Almost half of bedtime drink (4)
I believe the answer is:
nigh
'almost' is the definition.
('nigh' can be a synonym of 'almost')
'half of bedtime drink' is the wordplay.
'half of' means to take half the letters.
'bedtime drink' becomes 'nightcap' (I've seen this before).
'nightcap' cut in half is 'NIGH'.
(Other definitions for nigh that I've seen before include "'Almost, near (4)'" , "Near, as in archaic usage" , "Impending" , "See 23" , "Almost (arch.)" .)