A lady with a title shortly gets a man's name (4)
I believe the answer is:
adam
'a man's name' is the definition.
(Adam is a man's name)
'a lady with a title shortly' is the wordplay.
'lady with a title' becomes 'dame' (dame is a title).
'shortly' means to remove the last letter.
'dame' with its last letter removed is 'dam'.
'a'+'dam'='ADAM'
'gets' acts as a link.
(Other definitions for adam that I've seen before include "who transformed the Palladian style?" , "- - Bede, novel" , "father of us all?" , "Early man" , "First fruit-eater" .)