We investigate the stellar population properties of a sample of 24 massive quenched galaxies at 1.25 < zspec < 2.09 identified in the COSMOS field with our Subaru/Multi-object Infrared Camera and Spectrograph near-IR spectroscopic observations. Tracing the stellar population properties as close to their major formation epoch as possible, we try to put constraints on the star formation history, post-quenching evolution, and possible progenitor star-forming populations for such massive quenched galaxies. By using a set of Lick absorption line indices on a rest-frame optical composite spectrum, the average age, metallicity [Z/H], and α-to-iron element abundance ratio [α/Fe] are derived as log(age Gyr) 0.04-0.08+0.10, [Z/H] = 0.24-0.14+0.20, and [α/Fe] = 0.31-0.12+0.12, respectively. If our sample of quenched galaxies at 〈z〉 = 1.6 is evolved passively to z = 0, their stellar population properties will align in excellent agreement with local counterparts at similar stellar velocity dispersions, which qualifies them as progenitors of local massive early-type galaxies. Redshift evolution of stellar population ages in quenched galaxies combined with low redshift measurements from the literature suggests a formation redshift of zf ∼ 2.3, around which the bulk of stars in these galaxies have been formed. The measured [α/Fe] value indicates a star formation timescale of ≲1Gyr, which can be translated into a specific star formation rate of ≃1 Gyr-1 prior to quenching. Based on these findings, we discuss identifying possible progenitor star-forming galaxies at z ≃ 2.3. We identify normal star-forming galaxies, i.e., those on the star-forming main sequence, followed by a rapid quenching event, as likely precursors of the quenched galaxies at 〈z〉 = 1.6 presented here.