We present results from Chandra observations of the cluster lens SDSS J1029+2623 at z_l=0.58, which is a gravitationally lensed quasar with the largest known image separation. We clearly detect X-ray emission both from the lensing cluster and the three lensed quasar images. The cluster has an X-ray temperature of kT = 8.1 (+2.0, -1.2) keV and bolometric luminosity of L_X = 9.6e44 erg s^-1. Its surface brightness is centered near one of the brightest cluster galaxies, and it is elongated East-West. We identify a subpeak North-West of the main peak, which is suggestive of an ongoing merger. Even so, the X-ray mass inferred from the hydrostatic equilibrium assumption appears to be consistent with the lensing mass from the Einstein radius of the system. We find significant absorption in the soft X-ray spectrum of the faintest quasar image, which can be caused by an intervening material at either the lens or source redshift. The X-ray flux ratios between the quasar images (after correcting for absorption) are in reasonable agreement with those at optical and radio wavelengths, and all the flux ratios are inconsistent with those predicted by simple mass models. This implies that microlensing effect is not significant for this system and dark matter substructure is mainly responsible for the anomalous flux ratios. ; Comment: 35 pages, 8 figures. Accepted for publication in ApJ