The technique of michrobunched electron cooling (MBEC) is an attractive coherent cooling scheme with potential applications in future high-energy circular colliders. In our previous work, we analyzed the cooling of the energy spread using a one-dimensional (1D) technique that tracks the dynamics of microscopic fluctuations in the hadron and electron beams. In this paper, we extend this approach so that it covers the transverse emittance cooling as well. In order to do so, it is necessary to consider the betatron motion of the hadron beam and take into account effects of the momentum dispersion in the modulator and kicker regions. We derive relatively simple analytical expressions for the emittance and energy spread cooling times in terms of the various beam and lattice parameters, allowing us to perform fast optimization studies for an MBEC configuration. Verified through comparison with simulation, our theory can also incorporate features such as plasma amplification stages, which are crucial components of a realistic cooling system. Show Partial Abstract