Radar Receiver How to detect 1 nsec pulse
I'm not sure if your goal is to do the design your self or to just arrive at the functional unit within budget. Regardless here is a link to a pulse radar MMIC sensor that operates at 39 G samples/s. They offer multiple dev kits with some coming in under $300. You can choose to provide the software/MCU yourself or choose a kit where that is already done as well.

Xethru Radar Link