If my opponent plays Cull the Weak and my Ruin Runner is the only available unit, will I have to kill my runner or no since it's actually me choosing?

Yes, you have to kill your Ruin Runner.

Here's why: Cull the Weak does not target or choose any units when it's played. The instruction "Each player kills one of their units" is performed during resolution, and you are the one choosing which of your own units to kill (Rule 355.10.e). Ruin Runner says it "can't be chosen by enemy spells and abilities" — but since you (not an enemy spell) are the one making the choice at resolution, that protection doesn't apply. If Ruin Runner is your only unit when the spell resolves, you must choose it.