In this paper, we consider an intelligent reflecting surface (IRS) enabled mobile network, where a fixed access point (AP) communicates with a mobile user (MU) via the aid of an IRS. We assume that the MU moves from one elementary square to another following a Markov random walk within a grid, and propose a maximum a posteriori (MAP) criterion to track the movement of the MU by leveraging the line-of-sight component of the IRS-MU link. Since it is infeasible to derive an explicit expression for the average probability of estimation error (APEE) for the proposed MAP criterion, we derive a closed-form upper bound for the APEE, which is used as the cost function to optimize the phase shifts of the IRS units. Considering the unit modulus constraints incurred by the IRS units, a manifold optimization (MO) method is firstly employed to gain a favorable solution to the formulated optimization problem, followed by a low-complexity codebook based solution to circumvent the high computational cost of the MO method. Our numerical results demonstrate the superior performance of the proposed IRS phase shift designs over the benchmark method.