Does Arachnoid Horror allow me to play ambush units as reaction speed to a battlefield where there are only enemy units there?

No. Arachnoid Horror's permission and Ambush's timing do not combine that way.

Ambush (Rule 822.1.b) gives two things: permission to play to a battlefield where you control units, and [Reaction] timing only for that specific play location. The Reaction speed is conditional — it only applies when playing to a battlefield where you already have friendly units.

Arachnoid Horror adds a different permission: you may play to an occupied battlefield where an enemy is alone. But that doesn't give you [Reaction] timing on its own, and it doesn't satisfy Ambush's "where you control units" condition. Since you don't control any units at that battlefield, Ambush's [Reaction] timing is not available.

You could play the Ambush unit to that battlefield during your Main Phase at normal [Action] timing using Arachnoid Horror's permission — but not as a Reaction while the game state is Closed.