When Teemo, Strategist ability gets countered, does only 1 damage get countered or the entire ability itself?

The entire ability is countered — nothing happens at all: no reveal of the top 5, zero damage (not even 1), and no recycle.

A triggered ability is a single chain item, and countering acts on that whole item: "A card or ability that is Countered does nothing and is cleared from the chain" (Rule 426.1.a). There's no such thing as countering one instruction inside an ability. Even the "Deal 1 ... for each card with Hidden revealed this way" clause is one instruction inside that same chain item, so it can't be carved out separately.

The window matters too: your counter has to be played while Teemo's trigger is on the chain, before it starts resolving — you can't wait to see the reveals and then counter, since the reveal and the damage all happen inside resolution (FAQ #12523).