If I react to Annie readying her 2 runes at the end of turn, by playing Rengar into a battlefield, what happens with the showdown?

To answer your question, we must look at how the Ending Phase and Showdowns interact.

First, it is important to clarify a timing detail: Runes are readied during the Awaken Phase (at the start of a turn), not at the end of the turn. However, regardless of the timing, if you play Rengar, Trophy Hunter as a [Reaction] to an ability on the chain, the following sequence occurs based on the <showdown_flow>:

1. The Chain Resolves (LIFO)

• The Chain is Closed: When you play Rengar, Trophy Hunter, he is added to the top of the chain (he is a unit played as a reaction, so he enters the board at finalization per Rule 359.2.c).
• Resolution: Any items added to the chain after the original ability resolve first. Once those are finished, the original ability (the one you reacted to) resolves.
• Note: During this time, the state remains Closed. No Showdown can begin yet, because the turn must be in a Neutral Open State to trigger the start of a showdown (Rule 344).

2. Transition to Showdown

• Chain Empties: Once the chain is empty and both players pass focus, the game checks the state of the battlefields.
• Staging a Showdown: If your move resulted in you playing a unit to a battlefield controlled by your opponent (or an uncontrolled battlefield now occupied by both), the battlefield becomes Contested.
• Non-Combat vs. Combat:
  • If units from both players are present at the contested battlefield, a Combat Showdown is staged (Rule 344.1).
  • If only your units remain (e.g., if the opponent's unit left), a Non-Combat Showdown occurs (Rule 344.2).

3. Combat or Resolution

• If a Combat Showdown is staged, you proceed through the Steps of Combat (Combat Damage Step, then Combat Cleanup/Recall).
• If a Non-Combat Showdown is staged, the turn player initiates the showdown, and you establish control of the battlefield if only your units remain (Rule 348.2.a).

Summary

You cannot trigger a Showdown while an ability is still resolving on the chain. You must wait for the chain to fully empty. Once it empties, if the battlefield is Contested, the game will automatically transition into a Showdown (either Combat or Non-Combat) during the next cleanup or state check.

Disclaimer: This is my interpretation based on the mechanics of the Chain (LIFO resolution) and the rules for Showdowns. No official FAQ exists for this specific timing interaction.