Accurate channel state information (CSI) acquisition under tight pilot and training-energy constraints is essential for next-generation wireless networks. In this work, we model the wireless channel as a proper complex Gaussian process over the transmit and receive antenna arrays, reducing pilot overhead and training energy by estimating the CSI from partial observations. We formulate the CSI acquisition problem as a highly underdetermined Bayesian linear inverse problem. We develop a Gaussian process regression (GPR) framework that reconstructs the full CSI from sparse and noisy observations by extrapolating to the unknown entries. To incorporate propagation information into the GPR prior, we introduce a novel array-geometry-based kernel and prove that it is Hermitian positive semidefinite. The proposed kernel better captures the channel spatial correlations through richer hyperparameters. Our GPR-based CSI extrapolation approach learns the channel hyperparameters online from sparse, noisy pilot measurements within each coherence block. Numerical results show that the proposed estimator reduces pilot overhead by up to 75 percent and total training energy by up to 93.75 percent, while maintaining lower normalized mean-square error and higher spectral efficiency in the low-to-moderate signal-to-noise-ratio regime.