We use data from the Multi-Unit Spectroscopic Explorer (MUSE), recently commissioned at the Very Large Telescope (VLT), to study the kinematics and stellar population content of NGC 4371, an early-type massive barred galaxy in the core of the Virgo cluster. We integrate this study with a detailed structural analysis using imaging data from the Hubble and Spitzer Space Telescopes, which allows us to perform a thorough investigation of the physical properties of the galaxy. We show that the rotationally supported inner components in NGC 4371, i.e. an inner disc and a nuclear ring - which, according to the predominant scenario, are built with stars formed from gas brought to the inner region by the bar - are vastly dominated by stars older than 10 Gyr. Our results thus indicate that the formation of the bar occurred at a redshift of about z=1.8+0.5-0.4 (error bars are derived from 100 Monte Carlo realisations). NGC 4371 thus testifies to the robustness of bars. In addition, the mean stellar age of the portion of the major disc of the galaxy that is covered by our MUSE data is above 7 Gyr with a small contribution from younger stars. This suggests that the quenching of star formation in NGC 4371, which is very likely an environmental effect, was already occurring at a redshift of about z=0.8+0.2-0.1. Our results suggest that bar-driven secular evolution processes may have an extended impact on the evolution of galaxies, and thus on the properties of galaxies as observed today, and not necessarily be restricted to more recent cosmic epochs.