The Virgo Environmental Survey Tracing Ionised Gas Emission (VESTIGE) is a blind narrow-band Hα + [NII] imaging survey carried out with MegaCam at the Canada-France-Hawaii Telescope. During pilot observations taken in the spring of 2016 we observed NGC 4330, an intermediate mass (M∗ = 109.8 M⊙) edge-on star forming spiral currently falling into the core of the Virgo cluster. While previous Hα observations showed a clumpy complex of ionised gas knots outside the galaxy disc, new deep observations revealed a low surface brightness ∼10 kpc tail exhibiting a peculiar filamentary structure. The filaments are remarkably parallel to one another and clearly indicate the direction of motion of the galaxy in the Virgo potential. Motivated by the detection of these features which indicate ongoing gas stripping, we collected literature photometry in 15 bands from the far-UV to the far-IR and deep optical long-slit spectroscopy using the FORS2 instrument at the ESO Very Large Telescope. Using a newly developed Monte Carlo code that jointly fits spectroscopy and photometry, we reconstructed the star formation histories in apertures along the major axis of the galaxy. Our results have been validated against the output of CIGALE, a fitting code which has been previously used for similar studies. We found a clear outside-in gradient with radius of the time when the quenching event started: the outermost radii were stripped ∼500 Myr ago, while the stripping reached the inner 5 kpc from the centre in the last 100 Myr. Regions at even smaller radii are currently still forming stars fueled by the presence of HI and H2 gas. When compared to statistical studies of the quenching timescales in the local Universe we find that ram pressure stripping of the cold gas is an effective mechanism to reduce the transformation times for galaxies falling into massive clusters. Future systematic studies of all the active galaxies observed by VESTIGE in the Virgo cluster will extend these results to a robust statistical framework.