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We investigate the wave packet that remains bound in the ground and excited cationic states of oxygen after strong-field ionization by an intense 800-nm pulse. Much weaker probe pulses (800 or 264 nm) are used to dissociate these still-bound cations. The momentum distribution of O$^+$ is measured as a function of pump-probe delay and Fourier-transformed to obtain kinetic-energy-dependent and rotational-state-resolved quantum beat spectra. The sub-cm$^{-1}$ resolution of the Fourier transform allows unambiguous identification of the electronic, vibrational, and rotational states populated by the pump and then dissociated by the probe. Although strong-field ionization is expected to populate the lower-lying $X^2\Pi_g$ and $a^4\Pi_u$ states more effectively than the $b^4\Sigma^{-}_g$ state, a wave packet in the $X^2\Pi_g$ state is seen only with the 264-nm probe and only weak signatures of the $a^4\Pi_u$ states are found with either probe. The experiment confirms the role of the resonant coupling between the $b^4\Sigma^{-}_g$ and $a^4\Pi_u$ states by the 800 nm pulses [Xue \textit{et al.}, Phys. Rev. A 97, 043409 (2018)] and reveals the importance of rovibrational excitation in determining the momentum distribution of the O$^+$ fragments. The strong $X^2\Pi_g$ state contribution observed with the 264-nm probe also shows the importance of resonant coupling in the probe pulse. The sub-cm$^{-1}$ resolution also resolves spin-orbit splitting in both the $X^2\Pi_g$ and $a^4\Pi_u$ state wave packets.