If we were on the moon would earth appear stationary or would it appear to move. I think it must be stationary because moon is in sychronous rotation with earth.
Because the Moon is tidally locked the Earth will be in a nearly fixed place in the sky, while the sun rises and sets once every orbit (about once a month). There's a really cool animation/video from NASA that shows the moon undergoing libration as it orbits the Earth. From the point of view of the moon, the Earth would trace a path in the sky dictated by the libration motion of the moon.
You are correct, though there would be some small movement due to the fact that the Moon's orbit is not a perfect circle so the Moon is not in perfect synch.
The position of the Earth in the moon's sky wouldn't change much relative to the horizon if you didn't move around to different moon locations, but different locations would describe the altitude and azimuth of Earth differently. The changing radial position would cause the appearance of Earth to get larger or smaller.
Also, you would see the Earth rotate once every 24 hours and 50 minutes.
From the perspective of an observer on the Moon, the Earth will appear to move slightly over the course of a month and longer, for a number of reasons.
The Moon's rotational axis is tilted by about 6.7° with respect to the Moon's orbital plane. Just as the Earth's axial tilt causes us to see the Sun move from the north to the south and then back again over the course of a sidereal year, the Moon's axial tilt will result in the Earth appearing to move up and down by about 13 degrees over the course of a sidereal month.
The Moon's orbit is elliptical rather than circular. This means the Moon's orbital angular velocity varies over the course of its orbit about the Earth. This will make the Earth appear to move side to side by about 12 degrees over the course of an anomalistic month.
The Moon's orbit isn't quite elliptical due to perturbations from the Sun. These perturbations arise in a number of "anomalies", some of which such as the evection were known to the ancients, others of which such as were discovered by Brahe, Kepler, Newton, and others. These perturbations are more or less periodic, with periods ranging from a synodic month to a year to several years to tens of thousands of years. These effects are very small compared to the first two.
The first two effects completely dominate over the third, so I'll concentrate on them. These are the counter side to the Moon's libration as seen from the Earth. The Earth will appear to move back and forth across a diagonal line over the course of a month when the Earth's declination as seen from the Moon is at an extrema when the Moon as at perigee or apogee, close to a circle when the Earth crosses the Moon's equator when the Moon as at perigee or apogee, and an ellipse in between.