If I only have one rune up and an opponent kills my Ekko, can I let the death trigger go through to untap my runes and then play a Shen?

Yes, you can do this.

Here is the sequence of events:

1. Ekko dies: His Deathknell triggers and is placed on the chain as a pending item (Rule 808.1.d.2).
2. Priority: Before the Deathknell trigger resolves, you have an opportunity to act. You can exhaust your one ready rune to add energy to your pool (FAQ #9432).
3. Resolution: The Deathknell trigger resolves. You pay the mandatory cost (recycling Ekko) and the effect readies your runes (Rule 808.1.c, FAQ #1510).
4. Play Shen: Now that your runes are ready, you have priority again. You can play Shen as a [Reaction] (since he has the keyword) to a battlefield you control.

Important Notes:

• Control Requirement: As noted in FAQ #9449 and FAQ #8206, you can only play Shen to a battlefield you control. Ensure you are playing him to a location you control, not the battlefield where the combat is currently taking place if you are the attacker.
• Recycling is Mandatory: The "recycle me" portion of Ekko's Deathknell is a mandatory cost paid at resolution. You must be able to recycle Ekko to ready your runes (FAQ #1510).