Among currently available optical spectroscopic methods, Raman spectroscopy has versatile application to investigation of dynamical processes of molecules leading to chemical changes in the gas and liquid phases. However, it is still a challenge to realize an ideal standoff coherent Raman spectrometer with which both high temporal resolution and high-frequency resolution can be achieved, so that one can remotely probe chemical species in real time with high temporal resolution while monitoring the populations in their respective rovibronic levels in the frequency domain with sufficiently high spectral resolution. In the present study, we construct an air-laser-based Raman spectrometer, in which near-infrared femtosecond (fs) laser pulses at 800 nm and cavity-free picosecond N₂⁺ air-laser pulses at 391 nm generated by the filamentation induced by the fs laser pulses are simultaneously used, enabling us to generate a hybrid ps/fs laser source at a desired standoff position for standoff surveillance of chemical and biochemical species. With this prototype Raman spectrometer, we demonstrate that the temporal evolution of the electronic, vibrational, and rotational states of N₂⁺ and the coupling processes of the rovibrational wave packet of N₂ molecules can be probed.