Upcoming Irishman with degree coming in post (4-5)
I believe the answer is:
mail-order
'coming in post' is the definition.
(sent through the post)
'upcoming irishman with degree' is the wordplay.
'upcoming' says the letters should be written backwards (in a down clue coming up is backwards).
'irishman' becomes 'liam' (common Irish name).
'with' means one lot of letters go next to another.
'degree' becomes 'order' (order can mean a degree or rank).
'liam' written backwards gives 'mail'.
'mail'+'order'='MAIL-ORDER'
(Another definition for mail-order that I've seen is " what brides may be".)