Sauron's tears are for himself in this scene, not for Celebrimbor.
Sauron clearly has no concern for Celebrimbor's well-being, as is evidenced by the fact that he is torturing him in order to get him to reveal the location of the Rings. To Sauron, Celebrimbor was nothing but an instrument he used to achieve his plan of making the Rings of Power, and all of the deception as "Annatar" pretending to be Celebrimbor's friend was a means to that end. (That's in line with Charlie Vickers's comments about the scene.)
Sauron grows steadily more frustrated throughout the scene as he realizes that his usual manipulation tactics are no longer working. He tries the classic blame the victim approach: Why has Celebrimbor forced Sauron to torture him?
Look what you have done to yourself.
He threatens him with the violence the Orcs are likely to inflict on Celebrimbor when they arrive. Sauron promises a merciful death if Celebrimbor will give up the rings, and lasting torment if he will not.
He tries to appeal to Celebrimbor's ego:
All I have done has been for you. Yours was the fire that crafted the marvels of the age. I wish only to share them with the world. Please, let me.
But Celebrimbor is resolute:
But the Rings are beyond your reach. As I shall be, ere long. For soon I shall go to the shores of the morning. Borne hence, by a wind that you can never follow.
Then Celebrimbor delivers the line that makes Sauron lose his composure and kill the elven smith in a fit of rage, when he had until now been denying Celebrimbor the escape of death. It is a prophecy:
Shadow of Morgoth, hear the dying words of Celebrimbor! The Rings of Power shall destroy you. And in the end, I foresee, one alone shall prove your utter ruin!
Sauron thrusts a spear through Celebrimbor's chest and shouts
You're wrong. I am their creator. I am their master!
With his dying breath, Celebrimbor responds
No. You are their prisoner. Sauron, Lord... of the Rings.
This is when we see that Sauron is crying. He hurriedly regains his composure before the Orcs enter the room.
He is by no means sad that Celebrimbor is dead. He is likely furious that Celebrimbor outsmarted him by goading him into killing him. But what has truly shaken him is a threat to the one person Sauron holds in higher regard than anything else in the world: Sauron.
Celebrimbor's words have hit too close to the mark, as Sauron is in fact on the path to becoming a slave to the Rings even though his intention is to use them to enslave all others. The thought that this might be true is what has caused him to cry tears of anger. It is a thought that he cannot abide for even a moment, and that he will do anything to prove wrong.