The writer is personifying the building. The building, as if a person, is feeling anxieties, and the author is feeling some of them. The pat on the wall is an affectionate gesture of reassurance that you would give to a person.
EDIT: In reply to David's comment, it's possible that the statement was edited. I worked in video for years, and there were many times that we edited statements down to nearly-correct grammar that most people would understand and accept as they heard it. (Though this always bothered my boss, who was extremely detail-oriented.)
I hadn't noticed that you could click on the NPR and see the transcript in its entirety. The speaker mentions themselves, their team, etc, and uses several metaphors, like "taking an exam". So I think my interpretation is still quite reasonable, and to explain my reasoning in more detail:
"I start to feel some of the anxieties" indicates to me that the anxieties are not those of the speaker or any person or group. If it were the speaker's anxieties, he would say, "I start to feel some anxiety", and if it were the anxieties of other people, he would say, "I begin to feel their anxieties", or perhaps "I begin to experience our anxieties" (which is odd in its own way).
As I mentioned, it's possible that there was an audio edit and he originally said something like, "I start to feel some of the anxieties / that are a swirling around this project / and kind of give the facility a pat on the wall..."
The "pat on the wall for good luck" is odd. I guess it's possible that he's made his own ritual, but "pat" isn't an action I'd generally associate with trying to obtain luck. Common actions associated with obtaining good luck are: knocking on wood, rubbing a statue (Buddha's belly, a school mascot's nose), or in particular subcultures things like blowing (on dice in a casino) or slapping (a sign on the way out of a football locker room at the beginning of a game). A pat is generally comforting ("pat on the head"), encouraging ("pat on the shoulder"), congratulatory ("pat on the back"), etc.
So on the whole, it feels like the speaker has attributed the anxieties of those on the project to the building itself, and his gesture is one of reassuring the building that things will be okay, hence calming the building and in some sense decreasing the anxiety and increasing the chances of success.
It's possible that there were two or more edits or poor word choices and what the speaker meant was something like, "As I enter the building, I begin to feel anxiety about whether our project will ultimately succeed, and I pat the walls of the tunnel to give myself some reassurance that we built a firm foundation and will ultimately succeed."