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We present results from end-to-end simulations of observations designed to constrain the rate of change in the expansion history of the Universe using the redshift drift of the Lyman-$α$ forest absorption lines along the lines-of-sight toward bright quasars. For our simulations we take Lyman-$α$ forest lines extracted from Keck/HIRES spectra of bright quasars at $z>3$, and compare the results from these real quasar spectra with mock spectra generated via Monte Carlo realizations. We use the results of these simulations to assess the potential for a dedicated observatory to detect redshift drift, and quantify the telescope and spectrograph requirements for these observations. Relative to Liske et al. (2008), two main refinements in the current work are inclusion of quasars from more recent catalogs and consideration of a realistic observing strategy for a dedicated redshift drift experiment that maximizes $\dot{v}/σ_{\dot{v}}$. We find that using a dedicated facility and our designed observing plan, the redshift drift can be detected at $3σ$ significance in 15 years with a 25m telescope, given a spectrograph with long term stability with $R=50,000$ and 25% total system efficiency. To achieve this significance, the optimal number of targets is four quasars, with observing time weighted based upon $\dot{v}/σ_{\dot{v}}$ and object visibility. This optimized strategy leads to a 9% decrease in the telescope diameter or a 6% decrease in the required time to achieve the same S/N as for the idealized case of uniformly distributing time to the same quasars.