We investigate the stellar population properties of a sample of 24 massive quenched galaxies at $1.25<z_mathrm{spec}<2.09$ identified in the COSMOS field with our Subaru/MOIRCS 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 $alpha$-to-iron element abundance ratio [$alpha$/Fe] are derived as $log(mathrm{age}/mathrm{Gyr})=0.04_{-0.08}^{+0.10}$, $mathrm{[Z/H]}=0.24_{-0.14}^{+0.20}$, and $[alpha/mathrm{Fe}]=0.31_{-0.12}^{+0.12}$, respectively. If our sample of quenched galaxies at $langle z rangle = 1.6$ 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 $z_mathrm{f} sim 2.3$ around which the bulk of stars in these galaxies have been formed. The measured [$alpha$/Fe] value indicates a star formation timescale of $lesssim 1$ Gyr, which can be translated into a specific star formation rate of $simeq 1,mathrm{Gyr}^{-1}$ prior to quenching. Based on these findings, we discuss identifying possible progenitor star-forming galaxies at $z simeq 2.3$. 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 $langle z rangle = 1.6$ presented here.
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