How fast must you move in the direction of the sun for it to always be the same time of day?

Depends on your latitude, at the equator the Earth is around 40,000 km in circumference, so you'd need to travel 40,000 km in 24 hours, which is 1666 km/h. As you get further from the equator, the circle gets smaller and you don't need to go as fast until at the poles you just have to turn your head.

It varies with the latitude. But yes, up to a little over 1,000mph, to as slow as 0mph.

Depends on how close or far from the equator you were 😋

almost 1,000 mph

The speed of light

460/second

Oops

Very fast