How week begins with caviar for old president (6)
I believe the answer is:
monroe
'for old president' is the definition.
(James Monroe, US president 1817-1825)
'how week begins with caviar' is the wordplay.
'how week begins' becomes 'mon' (short for Monday).
'with' means one lot of letters go next to another.
'caviar' becomes 'roe' (fish eggs).
'mon'+'roe'='MONROE'
(Other definitions for monroe that I've seen before include "Marilyn ___ , film icon" , "Marilyn . . . . . ., tragic Hollywood icon" , "FILMSTAR" , "No more the blond film star" , "No more the old blond film start" .)