It's not that he will die within 1 year after the first time blood is sucked, he'll die within one year because he's gonna be killed by the other vampires (because he knows too much about vampires) and that turning into a vampire only works within one year of the first bite
I've been reading too much "oh im too powerful for the world, better to hide myself as a low-life nobody" that I'm starting to crave this kind of stuff
Inject this straight to my veins