The power per square metre at 100m is measured to be 4µW/m2. What would be a reasonable value to expect at 200m?

Power per square metre at 200 m = (4μW/m²) × (200/100) = 8μW/m²

This depends on WHAT THE HECK IS GOING ON.

If this power comes from a point source, then the inverse square law applies.
Since the sound or light or whatever spreads equally in all directions around a sphere, then at twice the radius the sphere will have 4 times more area, so the intensity will be 4 times less.
Makes sense doesn't it?

But there could be a thousand other factors involved.
Maybe you should post the WHOLE question next time?

Based on just the facts provided expect inverse square rule to apply,
so, double the distance, 1/4 the power