The activated ability can target only a tapped creature.

If the Oyster untaps during the draw step before its ability to add a -1/-1 counter resolves, the -1/-1 counter effect resolves and adds a -1/-1 counter after the -1/-1 counters are removed due to being untapped.

Giant Oyster’s ability can be activated during your upkeep, and the creature will get a -1/-1 counter during the draw step that follows it.

Activating the ability creates a continuous effect that works only as long as Giant Oyster remains tapped, a repeating delayed triggered ability that’s in effect only as long as Giant Oyster remains tapped, and a delayed triggered ability that triggers when Giant Oyster untaps or when it leaves the battlefield.