How do I make a specific area dark?

The problem is likely that either your skylight or sunlight don’t have “cast shadows” turned on, OR the roof of the cave does not cast a shadow. A roof can fail to cast a shadow if either it’s only turned inwards (single-sided triangles) with no separate outwards layer, OR if the mesh/asset in question is not marked as casting shadows.

If fixing those options doesn’t make it work for you, you’ll want to post a screen shot of the editor, somewhat zoomed out, showing the mouth/roof/innards of the cave, and the properties of that object opened in the property inspector, so someone else can perhaps figure out what’s going on.