On the Star Trek Blueprints - General Plans (1973) on deck 21 aft of the bowling alley there is a section that says
Seating Accommodation (83)
It is not labeled as such, but would I be wrong assuming it's a theater?
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With regard to the seating, it would appear that it faces directly onto the Enterprise's bowling alley which may or may not actually exist canonically, given that its sole mention (In TOS: "The Naked Time") was by a man who was temporarily insane.
You can see a depiction of the front couple of rows of seating in an easter egg from the Paramount-licensed video game Star Trek: Secret of Vulcan Fury
Yes, the seating shown on the right-hand side of that drawing is indeed a sort of "viewing gallery" for the ship's bowling alley. BTW, there was a bowling alley mentioned in an episode of TOS. In "The Naked Time," after he gets infected with that virus that strips its victims of their personal inhibitions, Ensign (or is he a Lieutenant?) O'Reilly takes control of Main Engineering and serenades the whole crew with an off-key rendition of the song "I'll Take You Home Again, Kathleen." In between verses of the song, O'Reilly at one point announces that there is to be a dance held in the ship's bowling alley. This, I think, is where Franz Joseph got the idea to include the bowling alley in his blueprints. So I think one can consider the existence of a bowling alley on Kirk's Enterprise as canon, since it was mentioned in an episode of the series.