Given a ground-level query image and a geo-referenced aerial image that covers the query's local surroundings, fine-grained cross-view localization aims to estimate the location of the ground camera inside the aerial image. Recent works have focused on developing advanced networks trained with accurate ground truth (GT) locations of ground images. However, the trained models always suffer a performance drop when applied to images in a new target area that differs from training. In most deployment scenarios, acquiring fine GT, i.e. accurate GT locations, for target-area images to re-train the network can be expensive and sometimes infeasible. In contrast, collecting images with coarse GT with errors of tens of meters is often easy. Motivated by this, our paper focuses on improving the performance of a trained model in a new target area by leveraging only the target-area images without fine GT. We propose a weakly supervised learning approach based on knowledge self-distillation. This approach uses predictions from a pre-trained model as pseudo GT to supervise a copy of itself. Our approach includes a mode-based pseudo GT generation for reducing uncertainty in pseudo GT and an outlier filtering method to remove unreliable pseudo GT. We validate our approach using two recent state-of-the-art models on two benchmarks. The results demonstrate that our approach consistently and considerably boosts the localization accuracy in the target area.