... is up to the first woman to sift ... (5)
I believe the answer is:
sieve
'sift' is the definition.
(I know that sift can be written as sieve)
'is up to the first woman' is the wordplay.
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'to the first' says to put letters next to each other.
'woman' becomes 'eve' (Eve is an example).
'is' in reverse letter order is 'si'.
'si'+'eve'='SIEVE'
'to' is the link.
(Other definitions for sieve that I've seen before include "Forgetful head is like it" , "Utensil for straining or sifting food" , "Screen, sift, strain" , "Jumblies' vessel" , "Fine strainer" .)