What is the subject of this sentence? the first in door bowling alley opened in 1840 in new york.

The subject of this sentence in particular is
"The first indoor bowling alley" because the entire sentence (or the predicate in this case) is based off of the subject of the first indoor bowling alley.

Although it might be confusing at first as to why I also mentioned "The first indoor bowling alley" instead of just "bowling alley," the answer is actually quite simple. The sentence doesn't talk about just any bowling alley but a specific one, and therefore "The first indoor bowling alley" would be the answer/subject of the sentence. :)