Does Astral Herron need to be at a battlefield the moment the first card is played, or only when you want to try to play the next card and use its effects' resources? I.e., I play a card earlier in the turn, resolve it, after which I move my Herron to a contested battlefield and want to use the discount that moment on an action/reaction card. Is this possible?

No. Heron must already be at a battlefield when your first card of the turn finishes resolving — the "if I'm at a battlefield" check happens at that moment, not later when you want to spend the discount.

The trigger is "When you play your first card each turn, if I'm at a battlefield…" — and that condition is evaluated right after the inciting event (your first card) fully resolves (Rule 383.2.c). If Heron is still in hand/on the board elsewhere at that instant, the conditional fails and the ability never triggers at all. There's no pending discount floating around that you can "activate" later by moving Heron (FAQ #12025).

So in your scenario: play card A, resolve it, then move Heron to a contested battlefield — the discount is gone. You'd have needed Heron at a battlefield before card A resolved. If you want the discount on an action/reaction played later in a showdown, you need to get Heron to a battlefield first and then play your first card of the turn while it's there.