How long would it take for a small plane to travel 1000 miles?

If your story airplane is jet powered the speed could go as fast as 500-600 mph but if you are sticking with turboprop then 200-250 mph.

Since it is futuristic story, you can as author make it as fast or slow as you need , artistic license :)

You need to be a lot more specific. Something like a single engine (prop) Cessna 172 cruises around 140 mph. In theory it could do it in 7.1 hours, but it only has a range of 800 miles, so it would have to stop for fuel which would add an hour or so. And that speed is airspeed, so the actual speed would depend on whether it has a headwind or tailwind.

The Gulfstream G650 (dual engine jet), which is still small compared to a commercial airliner can cruise at 90% the speed of sound (around 600 mph), so it could do it in about 2 hours (climb and descent are done at slower speeds).

You need to specify at least how many seats the plane will have. There are great variety of non-commercial aircraft.

Cessna 150 (two seater)

Top Speed: 122mph

Cruising speed: 117mph

Then it's about doing the math (though the plane will need to stop and land for gas) which is 8.5 hours.

If you tell me more specific type of aircraft you are looking toward using, I will edit my answer and find a plane of that size and its cruising speed.

4 hours @ 250 mph or 2 hours @ 500 mph