[Abridged]. Here, we report on the discovery of the galaxy counterpart of the z_abs=2.58 DLA on the line-of-sight to the z=3.07 quasar SDSS J091826.16+163609.0. The galaxy counterpart of the DLA is detected in the OIII 5007 and OII 3726,3729 emission lines redshifted into the NIR at an impact parameter of 16 kpc. Ly-alpha emission is not detected. The upper limit implies that Ly-alpha emission from this galaxy is suppressed by more than an order of magnitude. The DLA is amongst the most metal-rich DLAs studied so far at comparable redshifts. We find evidence for substantial depletion of refractory elements onto dust grains. Fitting the main metal line component of the DLA, which is located at z_abs=2.5832 and accounts for at least 85% of the total column density of low-ionisation species, we measure metal abundances from ZnII, SII, SiII, CrII, MnII, FeII and NiII of -0.12, -0.26, -0.46, -0.88, -0.92, -1.03 and -0.78, respectively. In addition, we detect absorption in the Lyman and Werner bands of hydrogen, which represents the first detection of H_2 molecules with X-shooter. The background quasar Q0918+1636 is amongst the reddest QSOs at redshifts 3.02<3.12 from the SDSS catalogue. Its UV to NIR spectrum is well fitted by a composite QSO spectrum reddened by SMC/LMC-like extinction curves at z_abs=2.58 with a significant amount of extinction given by A_V = 0.2 mag. This supports previous claims that there may be more metal-rich DLAs missing from current samples due to dust reddening of the background QSOs. The fact that there is evidence for dust both in the central emitting regions of the galaxy (as evidenced by the lack of Ly-alpha emission) and at an impact parameter of 16 kpc (as probed by the DLA) suggests that dust is widespread in this system.