In this paper, we propose and experimentally investigate a linearly polarized narrow-linewidth random fiber laser (RFL) operating at 1080 nm and boost the output power to kilowatt level with near-diffraction-limited beam quality using a master oscillation power amplifier. The RFL based on a half-opened cavity, which is composed of a linearly polarized narrow-linewidth fiber Bragg grating and a 500 m piece of polarization-maintained Ge-doped fiber, generates a 0.71 W seed laser with an 88 pm full width at half-maximum (FWHM) linewidth and a 22.5 dB polarization extinction ratio (PER) for power scaling. A two-stage fiber amplifier enhances the seed laser to the maximal 1.01 kW with a PER value of 17 dB and a beam quality of $M_x^2=1.15$ and $M_y^2=1.13$. No stimulated Brillouin scattering effect is observed at the ultimate power level, and the FWHM linewidth of the amplified random laser broadens linearly as a function of the output power with a coefficient of about $0.1237\mathrm{pm}/\mathrm{W}$. To the best of our knowledge, this is the first demonstration of a linearly polarized narrow-linewidth RFL with even kilowatt-level near-diffraction-limited output, and further performance scaling is ongoing.