It doesn't matter where the card is during your end step, as long as it was in a graveyard when you cast it or activated one of its abilities.

Laboratory Drudge's ability will trigger at the beginning of each end step whether you've cast a spell from a graveyard or activated an ability of a card in a graveyard that turn or not. You can respond to that ability and do one (or both) of the things to draw a card.