At what point do they actually start hearing cases?
They would only be officially a justice after the President issues a written commission and after the nominee takes 2 oaths of office.
Q: With the Senate vote, is Kagan now officially a justice?
No, the final step in the confirmation process involves President Obama issuing a written commission to his nominee, who then must take two oaths of office before assuming her official duties.
Which cases would that person hear? Is it only cases that were first appealed to the Supreme Court after some date? Or is there some other rule?
Basically, the new justice can hear all cases as there is no rule preventing it.
An article by the National Law Journal states:
But another little-known option is available to the eight-member court: It could hear the cases in February or March, say, and the ninth justice, whenever he or she joins the court this term, could vote after reading the briefs and reviewing the argument on tape or by reading the transcript.
Court experts agree that there is no ironclad command that prevents justices from voting on cases that were argued before their tenure began. The only thing preventing such a practice is long-standing tradition that weighs against participating in cases that were argued before a different set of justices. But in a new era with a new president who defies tradition daily, this one could also fall.