I would use light beam sensors for the start and stop signal input to a digital stopwatch. Small distance light beam sensors are very common and cheap but for bigger ( more than a few mm) distances you might have to dig through a parts catalogue for suitable components to make one.
You might be able to adapt a cheap digital stopwatch if you can get at the start / stop button contacts.
Or you could get a digital panel mounting timer and use that, just make sure it has a millisecond setting.
Of course, you could also build a timer/ stopwatch but it almost certainly would be more expensive than the ready made ones :wink: