We develop a bottom-up computational method for excited-state dynamics and time-resolved spectroscopy signals in molecular aggregates, on the basis of ab initio excited-state calculations. As an application, we consider the charge separation dynamics and pump-probe spectroscopy in the amorphous P3HT/PCBM blend. To simulate quantum dynamics and time-resolved spectroscopy, the model Hamiltonian for single-excitation and double-excitation manifolds was derived on the basis of fragment-based excited-state calculations within the GW approximation and the Bethe-Salpeter equation. After elucidating the energetics of the electron-hole separation and examining linear absorption spectrum, we investigated the quantum dynamics of exciton and charge carriers in comparison with the pump-probe transient absorption spectra. In particular, we introduced the pump-probe excited-state absorption (ESA) anisotropy as a spectroscopic signature of charge carrier dynamics after exciton dissociation. We found that the charge separation dynamics can be probed by the pump-probe ESA anisotropy dynamics after charge-transfer excitations. The present study provides the fundamental information for understanding the experimental spectroscopy signals, by elucidating the relationship between the excited states, the exciton and charge carrier dynamics, and time-resolved spectroscopy.