Rising out of bed, old woman's a cougar (4)
I believe the answer is:
puma
'a cougar' is the definition.
(I know that cougar can be written as puma)
'rising out of bed old woman's' is the wordplay.
'rising' says the letters should be written in reverse (in a down clue, letters go up).
'out of bed' becomes 'up' (synonyms).
'old woman' becomes 'ma' (both can mean someone's mother).
'up' backwards is 'pu'.
'pu'+'ma'='PUMA'
(Other definitions for puma that I've seen before include "Large mountain cat" , "US wild cat" , "International rugby player" , "Big cat" , "American wild cat" .)