Experimental demonstration of tunable temporal Goos-Hänchen shift (GHS) in synthetic discrete-time heterolattices with scalar and vector gauge potentials is reported. By using Heaviside-function modulation in two fiber loops, we create a sharp gauge-potential interface and observe temporal GHS for total internal reflection (TIR), which manifests as a time delay rather than a spatial shift. The TIR occurs as the incident mode falls into the band gap of transmitted region with band shifting by scalar and vector potential. We find that both scalar and vector potential codetermine GHS by controlling the decay (imaginary part) and oscillation (real part) of a penetrated evanescent wave, in stark contrast to traditional spatial GHS only determined by the decay factor. We also observe diverging characteristics of GHS at band-gap edges where evanescent-to-propagating wave transition occurs. GHS for frustrated total internal reflection (FTIR) by a finite-width evanescent barrier is also demonstrated, which shows saturation properties to the single-interface TIR case under infinite-width limit. Finally, we develop an accumulation measurement method using multiple TIRs to improve the precision for measuring even tinier GHS. The study initiates precise measurement of temporal GHS for discrete-time reflections, which may feature potential applications in precise time-delay control and measurement.