Since Jews don't believe in hell for unbelievers, what will happen to Satan? Will he be punished for eternity?
Jews don't believe in eternal hell - hell is time limited to 1 year (or 11 months).
But hell is not just a place of punishment, it's a place for a soul to be "cleaned" and prepared for it's next reincarnation. It is said the "fire" of hell is really the feeling of shame a person feels for his actions.
So placing Satan in hell would not make any sense in that light, since Satan is not a person (or soul) who needs punishment.
Additionally Satan is an angel of God who was commanded to act as he does. Unlike the Christian belief, Satan is not rebelling against God.
In Christianity evil comes from Satan and good from God. In Judaism evil is simply the absence of good - it does not come from any entity. God is the sole power, there are no other supernal powers.
"Satan" isn't an individual being opposed to God the way he is in Christianity. "The satan" is a job description; it's basically the angel who will act as prosecuting attorney when you come to be judged, and is the one God sends to test/prove people like Job. You see ha-satan in the midrash; for example, there is one about how when Avraham is on his way to offer Yitzchak up as a burnt offering, ha-satan keeps interfering -- by becoming a river and blocking the way, by placing doubts in Avraham's mind, by trying to turn Yitzchak against him, and so on. But the key point is that ha-satan acts at God's direction.