We present FUSE and HST STIS observations of the absorption-line system near the emission redshift of the radio-quiet, X-ray-bright quasar HE 0226-4110 (z = 0.495, V = 15.2). The spectra cover the rest-frame wavelength range 610-1150 angstrom, and we detect a wide range of ionization species, including four adjacent stages of oxygen: O (III-VI), which reveal a striking change in gas kinematics with ionization. Examinaton of the O (VI) lambda lambda 1031, 1037 doublet profiles reveals no evidence for partial coverage or unresolved saturated structure. O (III) is only detected in a narrow feature that is also traced by the H (I) and C (III) lines, suggesting that they arise in the same gas. Absorption at the same velocity is also present in other species (N (IV), O (IV-VI), S (IV), and possibly Ne (VIII)), but the kinematics differ from the O (III), implying production in separate gaseous phases. The combination of H (I), O (III), and C (III) information yields an estimate of both the photoionization parameter and the metallicity of the O (III)-bearing gas: [O/H] +0.12(-0.03)(+0.16), log U = -2.29(-0.23)(+0.02). We discuss two possible locations for the gas in this associated absorption-line system: the narrow emission line region of the quasar, and the halo of the quasar host galaxy. An additional narrow (and thus photoionized) component that is only detected in O (VI) appears 58 km s(-1) redward of the O (III)-bearing gas with -0.35 <= log U less than or similar to 0.02. Additional structure is detected in the associated absorber in the form of two broad components that only appear in moderate- to high-ionization species.