We analyzed images of comet 49P/Arend-Rigaux on 33 nights between 2012 January and May and obtained R-band lightcurves of the nucleus. Through usual phasing of the data, we found a double-peaked lightcurve having a synodic rotation period of 13.450+/-0.005 hr. Similarly, phase dispersion minimization and the Lomb-Scargle method both revealed rotation periods of 13.452 hr. Throughout the 2011/2012 apparition, the rotation period was found to increase by a small amount, consistent with a retrograde rotation of the nucleus. We also reanalyzed the publicly available data from the 1984/1985 apparition by applying the same techniques, finding a rotation period of 13.45+/-0.01 hr. Based on these findings, we show that the change in rotation period is less than 14 s per apparition. Furthermore, the amplitudes of the lightcurves from the two apparitions are comparable, to within reasonable errors, even though the viewing geometries differ, implying that we are seeing the comet at a similar sub-Earth latitude. We detected the presence of a short-term jet-like feature in 2012 March, which appears to have been created by a short-duration burst of activity on March 15. Production rates obtained in 2004/2005, along with reanalysis of the previous results from 1984/1985, imply a strong seasonal effect and a very steep fall-off after perihelion. This, in turn, implies that a single source region, rather than leakage from the entire nucleus, dominates activity.