There are two big ways that we think mortality plays a role in "Reading Moby-Dick at 30,000 Feet." First, the only reason we worry about wasting time is that we only have so much of it. Our speaker seems worried about wasting his life living passively, and not getting a chance to feel excitement, danger, and a connection to the world and to his feelings. At the same time, though, facing the risk of death (by doing something dangerous like whale hunting) seems to be almost part of the appeal of the fantasy life as a crewman on the Pequod for our speaker.
Death is seen from a strange perspective in this poem. The idea of dying at the end of a lifetime spent in "a room so large, / a corridor so long" is scary, whereas the possibility of drowning on a whaling voyage seems like part of the excitement. So, from our speaker's perspective, death is really only bad when it comes before you have a chance to live in the way he idealizes.
The speaker is off his rocker. Killing a big whale shouldn't make anyone feel more alive. It should make them feel really, really guilty.