Rising actor receives award that's primarily for play (7)
I believe the answer is:
macbeth
'for play' is the definition.
(Shakespeare play)
'rising actor receives award that's primarily' is the wordplay.
'rising' shows that the letters should be reversed in order (going up in a down clue).
'actor' becomes 'ham' (ham is a kind of actor).
'receives' is an insertion indicator (inserted letters are received).
'award' becomes 'CBE' (Commander of the Order of the British Empire award).
'primarily' suggests taking the first letters.
The initial letter of 'thats' is 't'.
'ham' back-to-front is 'mah'.
'cbe'+'t'='cbet'
'mah' going around 'cbet' is 'MACBETH'.
(Other definitions for macbeth that I've seen before include "That play about the Scottish king" , "its name must not be mentioned" , "from 6 & 12 VERDI & SHAKESPEARE] ?" , "play with tragic results" , "Shakespearean tragedy" .)