The Moon is inching away from the Earth every year and satellites require fuel to correct their orbital decay. The tidal force that the Sun and Moon would have on this mass would cause considerable perturbations and the thing would either descend to the Earth or get pulled further out into space. I suspect that the friction from the Earths atmosphere would cause the thing to come closer to us. At 1,000 mph they had better come up with some heat shielding.
There is no atmosphere 30,000 miles up, so no friction for orbital decay. The moon is receding from earth about 1.5 inches per year because it is not in geosynchronous orbit. The earth’s spin is faster than its orbital period, and tidal friction is pushing it away. Geosynchronous satellites do need to correct their orbits occasionally because of tidal forces of the moon and other objects, however. But if we can bring an asteroid into earth orbit, we should also be able to correct it occasionally. We’re talking about a small asteroid, not a flying mountain.