Occupation: Cowboy
The Cowboy is the archetypical character of the Western, perhaps the quintessential American hero. In the simplest terms, a "cowboy" is someone whose primary job is tending a herd of cattle on a ranch. In more general terms, it can be any character that has the appearance and mannerisms of a cowboy. Thus, the term "cowboy" is often used as an inclusive term for any Western characters, regardless of whether they are actually ranchers or not (TV Tropes, Cowboy).
In a lot of ways, the cowboy tends to be a very good shot or a loner, both in Billy's case. Although I use loner in the term that Billy is emotionally alone or lacks support not that he distances himself completely from others. I get the idea that he quite likes to be around people but just doesn't know how to commit to them emotionally. He's not looking for anything long standing, especially with woman. I feel this is in part to the guilt he carries for his mother's sacrifice. On the other hand, he knows that his behavior affects his work and he does take into consideration Soh and Hayate's rebukes. His companions are also understanding in his need to leave now and again choosing to believe that he's allowed to do with his free time as he chooses.
Through the interactions with the player character we find out that Billy owns a ranch in Texas and that he intends to go back to it. He invites the player character to go with him and through his story eventually marries her. I don't know if he has cattle or anything, but I assume as he settled down he'd get back into the swing of being more of a rancher and less of a sharp shooter, but the two might be so engrained in his personality that he wouldn't let his gun go. The idea that his clairvoyance is aided by his gun would also probably assist in his ranching.
He also fits the category of Gunslinger which often falls into the cowboy archetype. So, his role is simply that of the hero. And he looks cool while being a bit of a bad boy, but his heart is made of gold because he values hard work, honesty and he would never do anything to hurt anyone if he didn't absolutely have too. He's practical and to the point and he understands that he sometimes has to shoot, injure, or kill people to do his job.