What is the correct Islamic course of action if a friend confesses to a crime he committed, and then another innocent person is arrested for it?

You must advise your friend to confess his crime to prevent an innocent person from being unjustly punished. If he does not, then you must testify to what he confessed, or inform the innocent person. Testifying is obligatory if the innocent person requests it, especially if refraining from it would lead to the prescribed punishment being inflicted upon someone other than the perpetrator, or if a right depends on your testimony.