I can talk about the math in the abstract but since we have no figures here, it will be rather abstract.
The two factors will be:1. Absolute magnitude of the suns2. Distance from the sun to the surface when the sun is directly overhead.
The suns' light falls off by the square of the distance. This will determine its relative magnitude, which will increase until it passes over ahead, and decrease as it goes away. This will have to factor in the overhead distance. If the sun passes 3 (units) overhead, when it reaches 4 (units) farther away from that point, it will be only 5 (units) away from the person on the ground. Hence, if we measure give the brilliance of the sun at directly overhead a measurement of 100, it will be 36 when 5 units away -- 5 divided by 3, result squared, used to divide 100.
If it were alone, the new sun will first appear in the sky when the relative magnitude rises to a high enough level that a human eye can see it against the ambient light. The human eye is capable of seeing quite dim objects, so the actual factor is more likely to be that the prior sun is still putting out enough light to drown it out. (The luminosity difference between the Sun and the dimmest stars visible to the naked eye on a clear night past twilight and with no ambient light sources, either the moon or artificial, is about 10 to the 14th power.)
There would be no "night" vs. "day" You would have the sun overhead in its full brilliance, and then it would move off, slowly dimming, until it was dim enough that the new sun could be seen, and then it would continue to dim as the new sun brightened. The peak darkness would be the point at which the two suns were equal in brilliance. Then one would brighten as the other faded.
Distance from the surface will be important because that will decrease relative magnitude in a way that is not entirely dependent on the motion. There could be some very dark periods but the variation in light and dark would be continuous.
