Rotating machinery often operates under diverse conditions, resulting in significant domain shifts and nonstationary degradation patterns that pose challenges for robust fault diagnosis and remaining useful life (RUL) prediction. Existing models typically focus on either classification or regression tasks within a single domain, limiting their generalization capabilities. To address these issues, this paper proposes a unified multi-task learning framework based on a Mixture of Experts (MoE) architecture with domain adversarial training. The model integrates multiple specialized expert networks and a dynamic gating mechanism to extract discriminative features from various signal modalities, while concurrently performing fault classification and RUL regression. Experimental results on benchmark datasets demonstrate that the proposed method achieves superior performance in both diagnostic accuracy and RUL estimation robustness, especially under unseen working conditions. This work highlights the potential of combining modular sparse representation and adversarial domain adaptation to build scalable, transferable prognostic health management (PHM) models for industrial rotating machinery.