Sparse Matrix-Vector Multiplication (SpMV) performance on modern GPUs is frequently bottlenecked by the inherent mismatch between sparse data irregularity and regular hardware parallelism. Existing libraries typically rely on rigid ``one-size-fits-all'' configurations, struggling to adapt to diverse sparsity patterns. Moreover, they often neglect preprocessing efficiency, leading to severe overheads that negate kernel-level optimizations. To address these limitations, we introduce AtSpMV, an adaptive framework driven by a dual-stage performance model. It combines offline hardware profiling with a low-overhead online phase. By integrating the offline hardware model with the specific features of the incoming matrix, AtSpMV dynamically determines optimal parameters such as tile shape and load distribution. Furthermore, it features a GPU-accelerated, load-balanced preprocessing workflow to drastically reduce matrix preparation time. Comprehensive evaluations on NVIDIA A100 and H100 GPUs across 2,798 matrices demonstrate that AtSpMV achieves average speedups of 1.28x to 5.45x (up to 585x) over state-of-the-art libraries, including cuSPARSE, CSR5, DASP, and amgT. In real-world PageRank applications, AtSpMV delivers up to a 4.18x end-to-end speedup.