A few thoughts on this.
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The "Solo" system at the link says it work with "an incoming supersonic bullet", so I'm guessing it's sensing the shock wave as the bullet passes by, just before it hits the target. The sensors appear to have two microphone elements in each assembly. An AirSoft projectile isn't supersonic, so it presumably wouldn't work with that system.
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One approach might be to sense the shock wave as the projectile hits a solid target, e.g. a sheet of plywood. The speed of sound is considerably faster in wood than air (about 10x) so positional accuracy for any time measurement would be that much worse. Without knowing much about it, I'd be concerned about how uniform and predictable shock propagation in wood might be and I'd expect something like a canvas backstop to be even messier.
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That said, it wouldn't be difficult or expensive to attach a microphone or piezo sensor to a board and instrument it to see what happens. I'd start with a pair of those cheap KY-038 clap sensor microphones and watch the output with a digital oscilloscope or a USB logic analyzer.
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Most microcontrollers have counter/timer(s) with input capture that can capture the time at the instant of an external pulse, generally at a clock resolution up to the processor speed. For this project, one would like a counter/timer with multiple input capture channels on a single timer. I don't believe any of the common AVR Arduinos have that capability. I've worked with STM8 and STM32 microcontrollers which do have a timer with 4 capture channels. There are probably others. Assuming a propagation velocity of 3500 m/s and a 16 MHz counter gives the equivalent of sub-millimeter resolution which suggests time capture isn't going to be the issue. The limiting factors are likely to be sensor bandwidth and peculiarities of sound propagation in the target.