Supermassive black holes at the centres of galaxies can cycle through periods of activity and quiescence. Characterising the duty cycle of active galactic nuclei is crucial for understanding the impact of the energy they release on the host galaxy. For radio AGN, this can be done by identifying dying (remnant) and restarted radio galaxies from their radio spectral properties. Using the combination of images at 1400 MHz produced by Apertif, the new phased-array feed receiver installed on the Westerbork Synthesis Radio Telescope, and images at 150 MHz provided by LOFAR, we have derived resolved spectral index images (at a resolution of ~15 arcsec) for all the sources within ~6 deg^2 area of the Lockman Hole region. We were able to select 15 extended radio sources with emission (partly or entirely) characterised by extremely steep spectral indices (steeper than 1.2). These objects represent radio sources in the remnant or the restarted phases of their life cycle. Our findings suggest this cycle to be relatively fast. They also show a variety of properties relevant for modelling the evolution of radio galaxies. For example, the restarted activity can occur while the remnant structure from a previous phase of activity is still visible. This provides constraints on the duration of the 'off' (dying) phase. In extended remnants with ultra-steep spectra at low frequencies, the activity likely stopped a few hundred megayears ago, and they correspond to the older tail of the age distribution of radio galaxies, in agreement with simulations of radio source evolution. We find remnant radio sources with a variety of structures (from double-lobed to amorphous), suggesting different types of progenitors. The present work sets the stage for exploiting low-frequency spectral index studies of extended sources by taking advantage of the large areas common to the LOFAR and the Apertif surveys.