The class imbalance problem is a pervasive challenge in various applications, prompting researchers to design specialized loss functions for instance-level cost-sensitive learning models. However, the performance deteriorates when datasets are contaminated by label noise, especially in scenarios with uneven noise distribution. Additionally, specific hard-to-classify samples can further impede performance. Addressing these issues and establishing a robust instance-level cost-sensitive learning framework remains a formidable challenge. In this paper, we propose a robust Two-Stage instance-level cost-sensitive learning method with the Bounded Quadratic Type Squared Error (BQTSE) loss function, termed TSBQT. The asymmetric and bounded nature of the BQTSE loss function not only allows for distinct cost assignments to majority and minority instances, but also enhances the robustness to uneven noise. Moreover, our two-stage learning strategy prioritizes hard-to-classify instances by considering their classification difficulty. We solve TSBQT using the alternating direction method of multipliers and gradient descent algorithm. The generalization capability of TSBQT is analyzed with the Rademacher complexity theory. Extensive experiments validate the effectiveness of TSBQT.