The way a bullet chronograph works is it measures the momentary change in light intensity through an aperture with a diffuse bright background (either the sky, or a white plastic diffuser), by means of an infrared or visible-light photodetector such as a phototransistor or photodiode.
If you don't need a super accurate XY coordinate on the plane, you could potentially use the same method with an array of photodetectors along each axis, and some optics or shields to restrict which parts of the plane each photodetector can "see". You would then calculate the position of the projectile as it passes across the plane by looking at which photodetectors triggered (changed intensity) at the same time.
Alternatively, a very high speed CCD (or two) could work. You could calibrate the system using a reference plane printed on a panel, to know which pixel corresponds to which coordinate on the plane. I doubt the Arduino has enough processing power for running computer vision algorithms, so this may not be plausible in your application.