I bet that hurt a lot
Maybe, maybe not. This happened in 1897, and while the doctor-barber thing was still commonplace, this was essentially the golden age for anesthetics. Cocaine had been in use as a local anesthetic for a little over a decade, and morphine had been around (and more importantly, hadn't been made a strictly regulated drug at that point) for nearly a century. Additionally Laudanum would have been widely available, cheap, and wouldn't be *tightly* regulated until 1970.
I'll admit, I had to read that several times before it really sank in, and was even more surprised to find that it was available *over the counter* until the early 1950's.
All of that aside, this was a whaling vessel, rather than Navy, so it's altogether possible (and incredibly probable) that Captain Tilton just got the guy good and drunk before lopping his finger off.