In this passage, Jonah shows his faith in God in the midst of trouble. To this point, Jonah clearly believes in God but runs away from his clear calling instead of relying on God to do the task he feels ill-equipped, unable, or unwilling to do. It isn't until he has no other options, until his life bottoms out, that he calls upon God in faith for help. All too often, don't we do exactly the same thing? It isn't until we are "deep inside Sheol" (or more literally translated "in the belly of Hell/Hades/Sheol") that we give up trying to do things our way, concede that God's will is good, and reach out to him in faith. Woe to us for being such blockheads!
The parallels between Jonah and Jesus--explicitly pointed out by the latter--are not missed by the early church. Cyril of Jerusalem wrote a beautiful comparison in his beloved Catechetical Lectures. "When we study the story of Jonah the force of the resemblance becomes striking. Jesus was sent to preach repentance. So was Jonah. Though Jonah fled, not knowing what was to come, Jesus came willingly, to grant repentance for salvation. Jonah slumbered in the ship and was fast asleep amid the stormy sea; while Jesus by God’s will was sleeping, the sea was stirred up, for the purpose of manifesting thereafter the power of him who slept. They said to Jonah, 'What are you doing asleep? Rise up, call upon your God, that God may save us,' but the apostles say, 'Lord, save us!' In the first instance they said, 'Call upon your God,' and in the second, 'save us.' In the first Jonah said to them, 'Pick me up and throw me into the sea, that it may quiet down for you'; in the other Christ himself 'rebuked the wind and the sea, and there came a great calm.' Jonah was cast into the belly of a great fish, but Christ of his own will descended to the abode of the invisible fish of death. He went down of his own will to make death disgorge those it had swallowed up, according to the Scripture: 'I shall deliver them from the power of the nether world, and I shall redeem them from death' (Cyril, Catechetical Lectures).