We demonstrate tunable Schottky barrier height and record photo-responsivity in a new-concept device made of a single-layer CVD graphene transferred onto a matrix of nanotips patterned on n-type Si wafer. The original layout, where nano-sized graphene/Si heterojunctions alternate to graphene areas exposed to the electric field of the Si substrate, which acts both as diode cathode and transistor gate, results in a two-terminal barristor with single-bias control of the Schottky barrier. The nanotip patterning favors light absorption, and the enhancement of the electric field at the tip apex improves photo-charge separation and enables internal gain by impact ionization. These features render the device a photodetector with responsivity (3 A/W for white LED light at 3 mW/cm2 intensity) almost an order of magnitude higher than commercial photodiodes. We extensively characterize the voltage and the temperature dependence of the device parameters and prove that the multi-junction approach does not add extra-inhomogeneity to the Schottky barrier height distribution. This work represents a significant advance in the realization of graphene/Si Schottky devices for optoelectronic applications.