The authors introduce Flow Annealing Posterior Sampling (FAPS), a novel framework that unifies stochastic-process regression with PDE inverse problems in function space. Built upon pretrained function-space flow-matching priors, FAPS facilitates likelihood-guided posterior inference using sparse and noisy observations. The method supports variable query discretizations and avoids the need for explicit prior-density evaluation during sampling. It employs a Langevin correction mechanism that utilizes a low-rank covariance preconditioner to exploit dominant function-space correlations across different discretizations. Benchmarks on both Gaussian and non-Gaussian stochastic processes demonstrate that FAPS produces coherent posterior samples with accurate uncertainty quantification. The approach significantly outperforms existing functional regression baselines in these standard tasks. Furthermore, it achieves competitive or superior performance in noisy PDE inverse problems compared to diffusion-based samplers while reducing test-time sampling costs.