If the sacrificed creature was a Human, Falkenrath Aristocrat gains indestructible and also gets a +1/+1 counter.

You can sacrifice Falkenrath Aristocrat to pay the cost of its own ability. It won't receive a counter or gain indestructible, but it will enable abilities that check when or if a creature dies.

Falkenrath Aristocrat's activated ability checks whether the sacrificed creature was a Human as it last existed on the battlefield. It doesn't matter what its creature types are in the graveyard.