For thou hadst cast me into the deep, in the midst of the seas; and the floods compassed me about: all thy billows and thy waves passed over me. Then I said, I am cast out of thy sight; yet I will look again toward thy holy temple. The waters compassed me about, even to the soul: the depth closed me round about, the weeds were wrapped about my head.(Jonah 2:3-5) KJV
Jonah remembers sinking into the sea, and as he sank he saw the surface receding. He was mindful enough, while drowning, to think about the temple of the Lord in Jerusalem. Evidently he didn’t panic.
He was no doubt contemplating his imminent death and comforting himself with the thought that he would see the temple again in the resurrection. He was probably familiar with the book of Job.
It would take a while to sink to the bottom of the sea and get tangled up in seaweed, but even so it seems that Jonah didn’t freak out. He was reconciled to this outcome because he knew that he’d brought it upon himself. However, he believed that the Lord’s mercy was greater than his anger, and that when it was all over he would be in the presence of the Lord. He died, peacefully it would seem.