If I Temporal Breach my ready Thousand Tail Watcher, do I still pay the 1 cost and power to accelerate it or is the cost ignored?

No, the Accelerate cost is not ignored — you may still choose to pay it if you want.

Temporal Breach says "ignoring its cost," which sets Thousand-Tailed Watcher's base 7 Energy and 1 Mind Power to zero (Rule 356.1.b.1). Accelerate is an optional additional cost (Rule 805.2), not a base cost, so the "ignore" instruction doesn't touch it. Per Rule 356.1.b.3, additional costs still apply on top of zeroed base costs.

So you have two options:

• Don't pay Accelerate — Watcher enters for free but exhausted.
• Pay 1 Energy + 1 Mind Power — Watcher enters for free and ready.

(This matches the same logic as Baited Hook and other "ignore its cost" effects — see FAQ #10924 and #1293.)