Movable antenna (MA) has recently emerged as a promising paradigm for enhancing wireless communication performance by exploiting spatial degrees of freedom through flexible antenna repositioning. However, most existing designs rely on short-term user-specific instantaneous/statistical channel state information (CSI), which incurs excessive channel estimation overhead and complexity due to frequent antenna movement. To address this issue, this paper proposes a new design framework for antenna position optimization over a much longer timescale based on the cell-level statistical channel information acquired at the base station (BS). To this end, a cell-specific statistical channel model is developed for MA-aided multiuser communication systems, based on which the antenna position optimization framework for maximizing the ergodic system utility is formulated. Then, the covariance-eigenvalues-balancing antenna positions (CEBAP) design is derived to asymptotically approximate optimal solutions by statistically reducing users' channel correlation. Notably, the CEBAP solution solely depends on the BS-side angular power spectrum (APS) of wireless channels for mobile users across the cell, which significantly alleviates the overhead of channel acquisition and antenna movement, and yet remains effective for improving various system utilities over long timescales, such as weighted sum rate and minimum signal-to-interference-plus-noise ratio. Moreover, a low-complexity log-barrier penalized optimization (LOBPO) method is proposed to numerically solve the CEBAP. Simulation results based on realistic urban layouts and ray-tracing channels demonstrate consistent performance gains of the proposed CEBAP over fixed-position antenna systems across different utility functions, which closely approaches the upper bound achieved by instantaneous CSI-based MA optimization for moderately large antenna regions.