This paper presents a newly established sample of 19 unique galaxies and galaxy groups at redshift z = 0.8-1.21 in six QSO fields from the Cosmic Ultraviolet Baryon Survey (CUBS), designated as the CUBSz1 sample. In this sample, nine galaxies or galaxy groups show absorption features, while the other 10 systems exhibit 2 sigma upper limits of logN(HeI)/cm(-2) less than or similar to 13.5 and logN(OV)/cm(-2) less than or similar to 13.3. Environmental properties of the galaxies, including galaxy overdensities, the total stellar mass and gravitational potential summed over all neighbours, and the presence of local ionizing sources, are found to have a significant impact on the observed CGM absorption properties. Specifically, massive galaxies and galaxies in overdense regions exhibit a higher rate of incidence of absorption. The CGM absorption properties in galaxy groups appear to be driven by the galaxy closest to the QSO sightline, rather than by the most massive galaxy or by mass-weighted properties. We introduce a total projected gravitational potential psi, defined as -psi/G = Sigma(Mhalo)/d(proj) summed over all group members, to characterize the galaxy environment. This projected gravitational potential correlates linearly with the maximum density detected in each sightline (i.e. a power-law slope of 0.95(-0.14)(+0.15)), consistent with higher pressure gas being confined in deeper gravitational potential wells. In addition, we find that the radial profile of cool gas density exhibits a decline from the inner regions to the outskirts, and the amplitude is consistent with the cool gas being in pressure balance with the hot halo. Finally, we note that the ionizing flux from nearby galaxies can elevate the N(HI)/N(HeI) ratio, which provides a unique diagnostic of possible local sources contributing to the ionizing radiation field.