Fluid injection into underground formations reactivates preexisting geological discontinuities such as faults or fractures. In this work, we investigate the impact of injection rate ramp-up present in many standard injection protocols on the nucleation and potential arrest of dynamic slip along a planar pressurized fault. We assume a linear increasing function of injection rate with time, up to a given time t_c after which a maximum value Q_m is achieved. Under the assumption of negligible shear-induced dilatancy and impermeable host medium, we solve numerically the coupled hydro-mechanical model and explore the different slip regimes identified via scaling analysis. We show that in the limit when fluid diffusion time scale t_w is much larger than the ramp-up time scale t_c, slip on an ultimately stable fault is essentially driven by pressurization at constant rate. Vice versa, in the limit when t_c/t_w gg 1, the pressurization rate, quantified by the dimensionless ratio dfrac{Q_m t_w}{t_c Q^*} with Q^* being a characteristic injection rate scale, does impact both nucleation time and arrest distance of dynamic slip. Indeed, for a given initial fault loading condition and frictional weakening property, lower pressurization rates delay the nucleation of a finite-sized dynamic event and increase the corresponding run-out distance approximately proportional to propto left( dfrac{Q_m t_w}{t_c Q^*}right) ^{-0.472}. On critically stressed faults, instead, the ramp-up of injection rate activates quasi-static slip which quickly turn into a run-away dynamic rupture. Its nucleation time decreases non-linearly with increasing value of dfrac{Q_m t_w}{t_c Q^*} and it may precede (or not) the one associated with fault pressurization at constant rate only.