They don't get into details as to how the pre-cooler works but in the picture it looks like it might use pulsed airflow to achieve standing high and low pressure waves, and then tapping off air (or using a heat exchanger) at the low pressure/low temp points.

But the fundamental economical challenge is the need to expend the fuel to boost the whole thing to the upper atmosphere (or out of the atmosphere) where hypersonic speeds can be achieved at a reasonable fuel burn. The longer the leg, the easier it is to make the math work.