On the second coming of Jesus (Judgment), it is my understanding that God will fight and defeat Satan. What will happen to hell? Is there someone who will run it?
First of all, in a sense Satan has already been defeated. 1 Corinthians 15:26 says, "the last enemy that shall be destroyed is death." Not Satan, but death. Later in the same chapter (verses 55-57) Paul says, "O death, where is thy sting? O grave, where is thy victory? The sting of death is sin; and the strength of sin is the law. But thanks be to God, which giveth us the victory through our Lord Jesus Christ." Even though Satan still has power, we already know the victory belongs to God. In that sense, and in the sense that the power of sin has been defeated by God's grace and forgiveness, Satan has already been defeated. At the final judgement, though, his defeat will be final.
Hell was created for the devil and his angels (Matthew 25:41). Although souls of men will also be there, the real purpose was as a punishment for Satan. As such it will continue for eternity. There is no indication in scripture that Satan or anyone else will be in charge there. That idea is an invention of men. In fact, the nature of hell would make that unlikely. It is a place of darkness (Matthew 25:30), of utter absence of the light of God. It is inhabited by all kinds of wickedness, envy, and backbiting. As a place of utter darkness it would be unlikely that anyone could gain "control" of everything he can't see. As a place of evil it would be impossible for any one person to be in charge because total anarchy would exist. With no moral or fear restraints, everybody (or at least a large number of somebodies) would be trying to gain control, with nobody having an advantage. Add to that the utter pain and punishment. Nobody who would want to gain control will be able to try because they will be more worried about their own pain. As with anything that is devoid of God, chaos would exist, not organization.