If I have mutated Mouser at a battlefield, and my opponent plays Thousand-Tailed Watcher at base ready, applying a -3 to the mutated Mouser to 1 might, when the Thousand-Tailed Watcher comes to attack, how much might will the mutated Mouser have?

Your Mutated Mouser will have 1 Might.

According to the Thousand-Tailed Watcher ability and the supporting RiftJudge FAQ (#8931, #1448), the -3 Might reduction is a one-time modification applied when the ability resolves. It is not an ongoing static effect that continually checks or re-applies throughout the turn.

Sequence Breakdown

1. Watcher Played: Your opponent plays Thousand-Tailed Watcher. Its "When you play me" ability triggers and is placed on the chain.
2. Ability Resolves: The -3 Might reduction resolves. Your Mutated Mouser (base 1 Might) receives the -3 Might reduction. Because the effect includes a "minimum of 1" clause, your Mouser's Might becomes 1.
3. Snapshotting: Per Rule 472.3.d and FAQ #8931, the game "snapshots" this reduction for the remainder of the turn. The Watcher’s effect does not continue to check or reduce Might if it were to change later, nor does it create a permanent state that prevents your unit from having higher Might if you were to apply a buff.
4. Watcher Attacks: When the Thousand-Tailed Watcher eventually moves to attack, it does not re-trigger its "When you play me" ability. Your Mutated Mouser remains at the Might value established by the initial resolution of that ability.

Note on Interaction: If you were to play a buffing card (like a +1 Might spell) on your Mutated Mouser after the Watcher's ability resolved, your Mouser would increase normally. As established in FAQ #1448, Might modifications do not persist as ongoing effects, and subsequent buffs are simply added to the current value.