I understand that flying in either direction on a planet that has an atmosphere doesn't effect flight times, but on a body such as the moon, would it improve flight times?
Yes, your suborbital hop stays fixed in an inertial frame of reference, and the Moon rotate slowly underneath you.
It is not that important though, as the rotation rate of the surface of the Moon is only $4.6m/s$, relatively small compared to the orbital velocity of $1720m/s$
Actually, a similar but even weaker effect exists on Earth too, only eastwards instead. With the same velocity relative to the air, you are then moving at a higher fraction of orbital velocity, and it is then easier to counteract gravity.
The most efficient way of performing a flight on the Moon is a suborbital ellipse, with foci in the centre of the Moon, and directly between the two endpoints: