A hypothetical unit has "When you play your first card each turn, if I'm at a battlefield, your next card costs [2][Rune][Rune] less." If I play this card to a battlefield, does my next card get the discount? Assuming no other cards have been played this turn.

Yes. Your next card does get the discount.

Here's why. The trigger condition ("When you play your first card each turn") is evaluated after the inciting event — playing the card — has been fully processed (Rule 383.2.c). By that point, the unit has been finalized and has entered the board. So when the conditional "if I'm at a battlefield" is checked (Rule 383.3.e), the unit is already at the battlefield, and the trigger goes on the Chain. Rule 383.2.c.1 confirms this: a Game Object whose ability is active in a zone can evaluate its trigger if it enters that zone at the same time its condition is met.