If I kill Elder Dragon in a showdown, do I still die from his passive due to combat damage from Elder Dragon?

No, you will not take damage from Elder Dragon's passive ability, and you will not take standard combat damage from Elder Dragon if it is killed before the Combat Damage Step.

1. The Passive Ability

Elder Dragon’s ability ("Any amount of your damage is enough to kill enemy units") is a passive ability. A unit's passive abilities are only active while that unit is on the battlefield.

• Sequence: If you kill Elder Dragon during the Showdown window (which occurs during the combat phase), Elder Dragon is moved to the trash.
• Result: Because the unit is no longer on the board, its passive ability ceases to function immediately. When the combat proceeds to the Combat Damage Step, the ability is no longer active, so your units are not subject to its lethal threshold modification.

2. Standard Combat Damage

A unit that is killed before the Combat Damage Step cannot deal combat damage.

• Rule 460.2: The Combat Damage Step occurs after the Showdown closes.
• Rule 359.3.e.12 / FAQ 3573: The game follows the principle that a unit that leaves the board (is moved to the trash) is treated as having "null" attributes.
• Application: If Elder Dragon is killed during the Showdown, it is no longer present on the battlefield when the Combat Damage Step begins. Since it is not present to participate in the damage assignment process (Rule 460.2), it deals no combat damage to your units.

Summary of Timing

1. Showdown Window: You play a reaction to kill Elder Dragon.
2. Resolution: Elder Dragon is killed and moved to the trash.
3. Showdown Closes: Both players pass; the combat proceeds to the Combat Damage Step.
4. Damage Step: Elder Dragon is not on the battlefield. It does not deal standard combat damage, and its passive ability is not present to modify how other units deal damage.

Note: I could not fully verify this ruling against the retrieved rules.