Little is known of the activity and differential rotation of low luminosity, early M dwarfs from direct observation. We present the first stellar activity analysis of star-spots and faculae for the hot Jupiter hosting M1V dwarf Kepler-45 from Kepler transit light curves. We find star-spot and facula temperatures contrasting a few hundred degrees with the quiet photosphere, hence similar to other early M dwarfs having a convective envelope surrounding a radiative core. Star-spots are prominent close to the centre of the stellar disc, with faculae prominent towards the limbs, similar to what is observed for the Sun. Star-spot and facula mean sizes are about 40 and 45 × 103 km, respectively, and thus faculae occupy a 10 per cent larger surface area than the star-spots. A short-term activity cycle of about 295 d is observed that is reminiscent of those seen for other cool dwarfs. Adopting a solar-type differential rotation profile (faster equatorial rotation than polar rotation), our star-spot and facula temporal mapping indicates a rotation period of 15.520 ± 0.025 d at the transit latitude of -33.2°. From the mean stellar rotation of 15.762 d, we estimate a rotational shear of 0.031 ± 0.004 rad d-1, or a relative differential rotation of 7.8 ± 0.9 per cent. Kepler-45's surface rotational shear is thus consistent with observations and theoretical modelling of other early M dwarfs that indicate a shear of less than 0.045 rad d-1 and no less than 0.03 rad d-1 for stars with similar stellar rotation periods.