Fall recovery for legged robots remains challenging, particularly on complex terrains where traditional controllers fail due to incomplete terrain perception and uncertain interactions. We present \textbf{FR-Net}, a learning-based framework that enables quadrupedal robots to recover from arbitrary fall poses across diverse environments. Central to our approach is a Mass-Contact Predictor network that estimates the robot's mass distribution and contact states from limited sensory inputs, facilitating effective recovery strategies. Our carefully designed reward functions ensure safe recovery even on steep stairs without dangerous rolling motions common to existing methods. Trained entirely in simulation using privileged learning, our framework guides policy learning without requiring explicit terrain data during deployment. We demonstrate the generalization capabilities of \textbf{FR-Net} across different quadrupedal platforms in simulation and validate its performance through extensive real-world experiments on the Go2 robot in 10 challenging scenarios. Our results indicate that explicit mass-contact prediction is key to robust fall recovery, offering a promising direction for generalizable quadrupedal skills.