A part in that theatre of mine! (3)
I believe the answer is:
pit
This clue might be a double definition.
'a part in that theatre' is the first definition.
'pit' can be an answer for 'a part' (I have seen 'part of theatre' mean 'pit' so perhaps 'part' could also mean 'pit'). I'm not certain of the remainder of the definition.
'mine' is the second definition.
(down the pit)
'of' acts as a link.
Can you help me to learn more?
(Other definitions for pit that I've seen before include "Portland stone" , "Quarry" , "Match (one's wits); mine" , "Tip into hole" , "Sizeable hole in the ground" .)