Get a transit (or theodolite) and a GPS.

Get the program Stelarium (its freeware).

Use Stelarium to find some bright stars
(vega, polaris, Sirius (dog star is very bright in east).

Calibrate your transit so that Azimuth and Elevation angles are correct for Sirius (or other star of interest).

Then Track the moons Azimuth Elevation angles over the course of a month at 2-3 times a night.

Then you can check if the moon is out of it's normal place.