If you play Brynhir Thundersong on Frozen Fortress, then trigger Temporal Breach on your opponent's turn, does it get to play runes that turn?

Yes — the opponent still channels and uses their runes that turn. Brynhir only stops "playing cards"; channeling (Rule 430) is a game action, and rune abilities like "Add [1]" or "Recycle: Add [C]" are activations, not card plays — same logic as Brynhir not stopping gear abilities already on board (FAQ #8627).

Note the timing: her "When you play me" lockdown only covers "this turn," so replaying her via Temporal Breach on the opponent's turn locks down the rest of that turn — they can't play cards from hand or Hidden (including revealing Hidden cards), but runes and board abilities keep working. And since the opponent's Channel Phase happens at the start of their turn, they'll have already channeled their 2 runes before the lockdown even lands.