We present a multiplexed, high-resolution (R~50,000 median) spectroscopic survey designed to detect exoplanet candidates in two southern star clusters (NGC 2516 and NGC 2422) using the Michigan/Magellan Fiber System (M2FS) on the Magellan/Clay telescope at Las Campanas Observatory. With 128 available fibers in our observing mode, we are able to target every star in the core half-degree of each cluster that could plausibly be a solar-analog member. Our template-based spectral fits provide precise measurements of fundamental stellar properties - $T_{eff}$ ($\pm$30 K), [Fe/H] and [$\alpha$/Fe] ($\pm$0.02 dex), and $v_r\sin(i)$ ($\pm$0.3 km/s) - and radial velocities (RVs) by using telluric absorption features from 7160 to 7290 $\AA$ as a wavelength reference for 251 mid-F to mid-K stars (126 in NGC 2516 and 125 in NGC 2422) that comprise our survey. In each cluster we have obtained ~10-12 epochs of our targets. Using repeat observations of an RV standard star we show our approach can attain a single-epoch velocity precision of 25 m/s to 60 m/s over a broad range of S/N throughout our observational baseline of 1.1 years. Our technique is suitable for non-rapidly rotating stars cooler than mid-F. In this paper we describe our observational sample, analysis methodology, and present a detailed study of the attainable precision and measurement capabilities of our approach. Subsequent papers will provide results for stars observed in the target clusters, analyze our dataset of RV time-series for stellar jitter and stellar and sub-stellar companions, and consider the implications of our findings on the clusters themselves.
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