"Ulysses" is the Roman name for the Greek hero Odysseus, the mythical king of Ithaca who fought in the Trojan War alongside Achilles, Agamemnon, Menelaus, and others. He spent ten years fighting in the war, and then spent another ten years trying to get home! The Greek poet Homer wrote about Odysseus and the Trojan War in the Iliad, while his other epic poem the Odyssey is all about Odysseus's journey home.
Despite Odysseus' cunning – he's often referred to as "wily" or "crafty" Ulysses, and the whole Trojan Horse thing was his idea – it still takes him a while to combat all the monsters and near-death experiences the gods throw in his way.
In other versions of the story, Ulysses is in no rush to get home; in fact, he's willing to keep battling strange creatures and risk death in order to keep sailing. Both versions of the story of Ulysses are evoked in the poem; Ulysses is at home in Ithaca (Homer), but realizes he's bored and wants to keep exploring the world, even if it kills him (other versions). Getting home and resting up for a while is an important step, but so is getting back out there. Ulysses is like a football player who gets injured, goes to the locker room to get taped up, and opts go back in the game rather than spend the rest of it on the sidelines.