Quasar absorption systems not only affect the way quasars are selected, but also serve as key probes of galaxies, providing insight into their chemical evolution and interstellar medium (ISM). Recently, a method based on Gaia astrometric measurements has aided the selection of quasars reddened by dust hitherto overlooked. We conducted a spectroscopic study using VLT/X-Shooter on one such dust-reddened quasar, Q 2310─3358. This quasar, at z = 2.3909 ± 0.0022, is associated with a damped Lyman-α absorber (DLA) at nearly the same redshift 2.4007 ± 0.0003, with a neutral hydrogen column density of log N(H I) = 21.214 ± 0.003. The DLA is very metal-rich (close to solar metallicity after correction for depletion on dust grains). Its properties align with the metal-to-dust ratio and the mass-metallicity relation established in previous large samples of DLAs. Surprisingly, given its proximity to the quasar in redshift, the absorber has strong cold gas characteristics, including C I and H2. Based on the derived kinetic temperature of 71−15+28 71 − 15 + 28 K, we infer the presence of a strong UV radiation field, which in turn suggests that the quasar and the DLA are in close proximity, i.e., part of the same galaxy and not just different objects in the same overdensity of galaxies. We used the line ratios of the C I fine-structure lines to constrain the density of the cold gas, yielding nH ∼ 103 cm−3. Our analysis extends the understanding of zabs ≍ zem absorption line systems and provides valuable constraints on the interplay between dust, metals, and neutral gas in the ISM of early galaxies.