Enemy plays Astral Heron from hand to battlefield. Does the next card played by him get discounted?

Yes — assuming Astral Heron is the first card your opponent plays that turn and it resolves directly to a battlefield, it sees its own play and triggers, so their next card that turn costs [2][A][A] less.

The "if I'm at a battlefield" clause is checked when the trigger condition is met, which is when their first card of the turn finishes resolving — Heron is already sitting at the battlefield at that point, so it qualifies (FAQ #13157). Order of play matters though: they must let the Heron trigger resolve before playing the discounted card, and the discount expires at end of turn if unused (CR 420.4.a).

Two ways it fails: if they play a card before Heron that turn, the "first card" window has already passed and nothing triggers (FAQ #12289); and if they play Heron to their Base instead of a battlefield, the battlefield check fails and it never triggers retroactively if they move it later.