This just now occurred to me, and I can't believe nobody else has mentioned it yet (as far as I could find). The reason Richard Alpert knew that Daniel wasn't going to shoot him was because he knew that he still had yet to meet John Locke in the future! Locke told him in 1954 that they would meet in sometime in the future. Since that hasn't happened to Richard yet as of 1977, he knows that he will live at least long enough to meet Locke again, so that's why he's not worried when his life is threatened by Daniel pointing a gun at him. Now, what he didn't know is that the reason Daniel wasn't going to shoot him was because Eloise would shoot Daniel first! So essentially, Eloise unknowingly fulfilled a prophecy by saving Richard's life!