The strong sense of urgency that Ulysses radiates stems largely from his own consciousness of death; it seems like every time he talks about going back to sea he mentions the fact that he might die soon, or die out there. Ulysses knows that death is stalking him, and he wants to try and cheat it for as long as he can. In other words, he wants to try to steal as many moments as he can before the curtain drops. And he thinks by traveling more he can somehow forestall death, can make the "eternal silence" wait just a bit longer for him.
Ulysses realizes he might die, but he's not upset about it because he knows his fame will outlive his body.
Ulysses refers to the afterlife, but is ultimately unsure whether or not he'll go there and whether or not it even exists.