How long can a bed bug live with one drop of blood?

Depends on the sort. Newly hatched bedbugs are called nymphs and they need to have had five moults and five different meals of somebody's blood to become adults. If they do not get these within a week or two nymphs may well not survive.

But adult bedbugs can survive for a year or more between feeds. So just one extraction of your blood while you were asleep on one night is enough to enable that bedbug to remain alive for over a year after that meal. But in practice once that bedbug has enjoyed that one meal off you in that bed it is far more likely it will come back to enjoy more of your delicious blood every few nights thereafter giving you more itchy marks on your limbs, until you have found it and killed it.