As opposed to @TheLethalCarrot's answer, I would say no. There is no evidence that Bran can see into the future. Bran has been seen to have visions of the future, but not the active ability to look into the future.
During the council of the Battle of Winterfell, Arya asks "Dragon fire will stop him [the Night King]?" and Bran responds, "I don't know. No one's ever tried." We know Bran can actively see into the past and see anything happening in the present, but if he could actively see into the future, he would have seen Daenerys try to kill the Night King with Drogon.
So more likely, he did not know what would happen. He was calm because, as the Three Eyed Raven, his emotions have been dulled, if not completely nullified. In 'Inside the Episode' after the episode, when David Benioff says "Bran knows what is going to happen," I believe he simply meant they both knew that Theon was going to die. Obviously, as he can not hope to kill the Night King alone.
Just to address the other answers and the possibility that Bran did see and know the outcome, it is entirely possible but in my view, very unlikely. The fact that he gave Arya the Cat's Paw dagger isn't really indicative of knowing she would use it to kill the Night King. It could be just as he said, "It's wasted on a cripple" and he knows Arya is a badass now, which seems more likely. And who else would he give it to? Jon, who already has a valyrian steel sword? Jaime, who pushed him out the window and only has one hand? Brienne, a person he doesn't know? Arya was the only person he knew who he could really even give it to, and the best suited for a valyrian steel dagger.
Also, for the argument that Bran is keeping information from other people so not to prevent said future from happening also seems unlikely. As we learned from Hodor's backstory, he became Hodor because of Bran's meddling in the past. But Hodor was already Hodor before Bran became the Three Eyed Raven. This suggests the world of Game of Thrones follows a fixed timeline and not a multiverse like the MCU. It seems unlikely to me that Bran would think his actions could make a difference to his future visions considering what he has seen.
It seem's more likely to me that if he were to, for example, see Jon die in his vision, he would go up to Jon and say "You're going to die tomorrow. Love you bud." and roll away, although I'm not saying that's definitely true.
All in all, while it is entirely possible that Bran can and did see into the future and knew what would happen, there's no evidence to suggest that that is what happened. While it's completely possible Bran gave Arya the Cat's Paw dagger specifically to kill the Night King, until we have more information from the show or the creators/writers that that is true, we have to assume it is not.