If you play Astral Heron to a battlefield, how many cards would you have to play to get its effect?

One. You play Astral Heron as your first card of the turn. After it's finalized on the battlefield, its triggered ability checks the condition — it sees that you've played your first card and that Astral Heron is at a battlefield, so the trigger fires. When it resolves, your next card (the second card you play that turn) costs [2][Rune][Rune] less.

This is covered by FAQ #11610 (current): the trigger is evaluated after the inciting event is fully processed, so Astral Heron being on the battlefield during the "if I'm at a battlefield" check. You play one card to get the discount on the next.