Exact Kriging and conditional simulation (CS) for uncertainty quantification are computationally infeasible for modern spatial analyses with large numbers of observations and dense prediction grids. We present a rapid approximation to the Kriging prediction step for stationary Gaussian processes for a regular prediction grid by approximating each off-grid covariance vector by a sparse linear combination of on-grid covariances within a local $L$-order neighborhood of $M = (2L)^2$ neighboring grid points. This reformulation reduces complexity from $O(N n^3)$ to $O(N \log N + nM + M^3)$ while preserving accuracy. A factorial study shows that approximation error decreases systematically with increased Matérn smoothness, neighbor order $L$, and grid resolution, aligning with bounds from kernel approximation theory. In a North American summer-rainfall application ($n=1368$), our method produces predictions visually indistinguishable from exact Kriging with point-wise errors on the order of $10^{-5}$ inches and achieves more than $150$ times speedups at a $350\times350$ grid, also outperforming Vecchia and LatticeKrig predictions. Embedded in a fast CS scheme, the approach reproduces Kriging standard errors and scales favorably with both $n$ and $N$. We recommend a practical workflow that uses a fast method for parameter estimation followed by our rapid predictor for fine-grid mapping and uncertainty quantification.

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