Your first "if" is problematic. A full Moon occurs when the Moon is almost exactly opposite the Sun in the sky. As a result, if it is daytime at a given spot, you won't see the Moon -- if it is full, because it will be below the horizon. Notice I said "exact" -- it is possible, depending on the circumstances, for some locations (extreme northerly or southerly latitudes particularly) for a full Moon to appear just on the (opposite) horizon as the Sun is rising or setting -- this is from the "almost exactly opposite" disclaimer that I mentioned above.

If, alternatively, the Moon is at the first quarter phase, then it will rise, roughly at local noon, and set at local midnight. If one were to look out in Virginia at 3 p.m. when the moon was first quarter, the Sun would be up -- but rising in the east would be the Moon. Meanwhile, on the opposite side of the Earth, both the Sun and the Moon would be below the horizon, because there the Moon *also* would be rising at local noon and setting at local midnight, but it would be 3 a.m., so the Moon would have already set.