when is the next ending phase for Runeblade Anvil? Is it at the end of my current turn or the end of my next turn?

Yes — it's the end of your current turn, so the banished card has to be played before you end this turn (assuming you just played Runeblade Anvil on your own turn).

"When you play this… banish the top card of your Main Deck. Until your next Ending Phase, you may play that card" — your next Ending Phase is simply the next one that belongs to you, and since the Anvil entered during your own Main Phase, that's the Ending Phase of this turn. The permission ends as that phase begins (Rule 317), and there's no discretionary window in the Ending Phase to play a card anyway, so your real last chance is your Main Phase or any Showdown this turn.

The one case where it's "end of your next turn" is if the Anvil leaves the board (including via its own [2][C], [T]: Kill this ability) during an opponent's turn — then your next Ending Phase is a full turn away, and you can play the card at any legal point during your next turn. Reminder that you still pay the card's costs, and it must be legally timed: Main Phase while you have priority, or Action/Reaction during a Showdown.