The one thing in favor of this scheme is the unequal distribution of mass in the Moon keeps one side always facing the Earth. But it does not stay over the same point on the Earth, so a physical connection would not be possible or even desirable.
The solution is to anchor a rope to the surface of the Moon, but cut it a few feet short of the distance to the Earth's surface. The Earth end of the rope can move freely as the Moon orbits. Then, when you want to get to the Moon without a rocket, you just need to grab the end of the rope as it goes by.