We present the first results from a new, deep (200 ks) Chandra observation of the X-ray luminous galaxy cluster surrounding the powerful (L similar to 10(47) erg s(-1)), high-redshift (z = 1.067), compact-steep-spectrum radio-loud quasar 3C 186. The diffuse X-ray emission from the cluster has a roughly ellipsoidal shape and extends out to radii of at least similar to 60 arcsec (similar to 500 kpc). The centroid of the diffuse X-ray emission is offset by 0.68 +/- 0 ''.11 (similar to 5.5 +/- 0.9 kpc) from the position of the quasar. We measure a cluster mass within the radius at which the mean enclosed density is 2500 times the critical density, r(2500) = 283(-13)(+18) kpc, of 1.02(-0.14)(+0.21) x 10(14) M-circle dot . The gas-mass fraction within this radius is f(gas) = 0.129(-0.016)(+0.015). This value is consistent with measurements at lower redshifts and implies minimal evolution in the f(gas)(z) relation for hot, massive clusters at 0 < z < 1.1. The measured metal abundance of 0.42(-0.07)(+0.08) Solar is consistent with the abundance observed in other massive, high-redshift clusters. The spatially resolved temperature profile for the cluster shows a drop in temperature, from kT similar to 8 keV to kT similar to 3 keV, in its central regions that is characteristic of cooling-core clusters. This is the first spectroscopic identification of a cooling-core cluster at z > 1. We measure cooling times for the X-ray emitting gas at radii of 50 kpc and 25 kpc of 1.7 +/- 0.2 x 10(9) years and 7.5 +/- 2.6 x 10(8) years, as well as a nominal cooling rate (in the absence of heating) of 400 +/- 190 M-circle dot year(-1) within the central 100 kpc. In principle, the cooling gas can supply enough fuel to support the growth of the supermassive black hole and to power the luminous quasar. The radiative power of the quasar exceeds by a factor of 10 the kinematic power of the central radio source, suggesting that radiative heating may be important at intermittent intervals in cluster cores.