Why was he crying? Annatar manipulates and kills easily, so why was he borderline sobbing after killing Celebrimbor?
Because this death was brutal, and more than that, it was personal. Sauron doesn't kill because he's a sadist, he kills because he is ruthless and believes it necessary to achieve his goals. Anyone who gets in his way, or who is more useful dead, will simply be so. But as he's torturing Celebrimbor for information about the rings, Celebrimbor is reading him to filth. And with compassion, even.
There is a spark of anger. Absolute rage within Annatar, and in his fury he kills Celebrimbor. Not because it was useful or the smart thing to do, because he was offended, even hurt by his words. He killed Celebrimbor in anger, which is not like him, and when he steps back and realizes how violently and twistedly he did it, to man he even respected and may have liked to some extent, something clicks.
Why was he crying?
Because he looked at what he'd done to Celebrimbor, and he saw Melkor. This was Morgoth's work. This was how he'd handled things, how he'd tormented and punished Sauron. All this time, as we know from his earlier conversation with Celebrimbor, Sauron believes he is being merciful to those he tortures and kills compared to Morgoth.
But in that moment, he has brief terrifying glimpse of self-awareness. He is the monster he feared and hated and thought he loved. He steps backs and looks at a scene Morgoth would paint and it terrifies him.
So he cries. He cries until the reality of his actions, and another opportunity walks through the door in the form of the orcs and he makes the same call he will always make: double down, justify it, make it worth the pain.
