We present Chandra studies of the X-ray binary (XRB) populations in the bulge and ring regions of the ring galaxy NGC 1291. We detect 169 X-ray point sources in the galaxy, 75 in the bulge and 71 in the ring, utilizing the four available Chandra observations totaling an effective exposure of 179 ks. We report photometric properties of these sources in a point-source catalog. There are {approx}40% of the bulge sources and {approx}25% of the ring sources showing >3{sigma} long-term variability in their X-ray count rate. The X-ray colors suggest that a significant fraction of the bulge ({approx}75%) and ring ({approx}65%) sources are likely low-mass X-ray binaries (LMXBs). The spectra of the nuclear source indicate that it is a low-luminosity active galactic nucleus (AGN) with moderate obscuration; spectral variability is observed between individual observations. We construct 0.3-8.0 keV X-ray luminosity functions (XLFs) for the bulge and ring XRB populations, taking into account the detection incompleteness and background AGN contamination. We reach 90% completeness limits of {approx}1.5x10^37^ and {approx}2.2x10^37^ erg/s for the bulge and ring populations, respectively. Both XLFs can be fit with a broken power-law model, and the shapes are consistent with those expected for populations dominated by LMXBs. We perform detailed population synthesis modeling of the XRB populations in NGC 1291, which suggests that the observed combined XLF is dominated by an old LMXB population. We compare the bulge and ring XRB populations, and argue that the ring XRBs are associated with a younger stellar population than the bulge sources, based on the relative overdensity of X-ray sources in the ring, the generally harder X-ray color of the ring sources, the overabundance of luminous sources in the combined XLF, and the flatter shape of the ring XLF.