Many a man gets this woman's address (5)
I believe the answer is:
madam
'this woman's address' is the definition.
(respectful term of address for a woman)
'many a man' is the wordplay.
'many' becomes 'M' (Roman numeral for a thousand).
'a man' becomes 'adam' (man's name).
'm'+'adam'='MADAM'
'gets' acts as a link.
(Other definitions for madam that I've seen before include "Stuck-up girl" , "Courteous form of address to a woman" , "Term of address to a woman" , "Formal address to a woman - precocious little girl" , "Conceited girl (colloq.)" .)