This paper presents a newly established sample of galaxies and galaxy groups at redshift z ≈ 1 from the Cosmic Ultraviolet Baryon Survey (CUBS), for which sensitive constraints can be placed on the circumgalactic medium (CGM) using high-quality far-ultraviolet (FUV) and optical absorption spectra of background QSOs. The CUBS program has uncovered 19 unique galaxies or galaxy groups at z = 0.89 to 1.21 in six QSO fields, which is designated as the CUBSz1 sample. In this CUBSz1 sample, nine galaxies or galaxy groups show absorption features, while ten systems do not have detectable absorption with 2-σ upper limits of log N(He i) /cm−2 ≲ 13.5 and log N(O v)/cm−2 ≲ 13.3. Environmental properties of the galaxies, including galaxy overdensities, the total stellar mass and gravitational potential summed over all nearby neighbors, 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. At the same time, the observed 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 ψ, defined as −ψ/G = Σ Mhalo/dproj summed over all group members, to characterize the overall 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 general 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 generate an elevated N(H i)/N(He i) ratio, which in turn provides a unique diagnostic of possible local sources contributing to the ionizing radiation field.