We report new Northern Extended Millimeter Array observations of the C II158 μm, N II205 μm, and O I146 μm atomic fine structure lines (FSLs) and dust continuum emission of J1148+5251, a z = 6.42 quasar, which probe the physical properties of its interstellar medium (ISM). The radially averaged C II158 μm and dust continuum emission have similar extensions (up to $\theta ={2.51}_{-0.25}^{+0.46}\ \mathrm{arcsec}$ , corresponding to $r={9.8}_{-2.1}^{+3.3}\ \mathrm{kpc}$ , accounting for beam convolution), confirming that J1148+5251 is the quasar with the largest C II158 μm-emitting reservoir known at these epochs. Moreover, if the C II158 μm emission is examined only along its NE-SW axis, a significant excess (>5.8σ) of C II158 μm emission (with respect to the dust) is detected. The new wide-bandwidth observations enable us to accurately constrain the continuum emission, and do not statistically require the presence of broad C II158 μm line wings that were reported in previous studies. We also report the first detection of the O I146 μm and (tentatively) N II205 μm emission lines in J1148+5251. Using FSL ratios of the C II158 μm, N II205 μm, O I146 μm, and previously measured C I369 μm emission lines, we show that J1148+5251 has similar ISM conditions compared to lower-redshift (ultra)luminous infrared galaxies. CLOUDY modeling of the FSL ratios excludes X-ray-dominated regions and favors photodissociation regions as the origin of the FSL emission. We find that a high radiation field (103.5-4.5 G 0), a high gas density (n ≃ 103.5-4.5 cm-3), and an H I column density of 1023 cm-2 reproduce the observed FSL ratios well.