Can you not activate Thousand Tail Watchers effect on one of your units so fast?

No, you cannot use Not So Fast to counter the "When you play me" ability of Thousand-Tailed Watcher.

According to the official ruling (FAQ #8449 and #8512):

• Not So Fast requires the enemy spell or ability to "choose" (target) a specific friendly unit or gear.
• Thousand-Tailed Watcher's ability applies a blanket effect to all enemy units simultaneously. Because it does not target or choose specific units, it does not meet the requirements for Not So Fast to be played against it.

While you cannot counter the ability with Not So Fast, you can still play other [Reaction] speed cards in response to the Watcher's "When you play me" trigger before the -3 Might penalty resolves.