While cancer survival analysis through multi-modal integration of histopathology imaging and transcriptomic profiling provides comprehensive prognostic insights, transcriptomic profiling remains technically demanding and cost-prohibitive in clinic practice. However, existing multi-modal survival analysis frameworks have critical dependence on complete data availability during clinical deployment, rendering them operationally impractical given the incompleteness of real-world patient datasets. Considering the clinical reality of incomplete modality availability, implementing cross-modal knowledge fusion during training offers a viable solution to this dilemma, enabling the deployed histopathology-based model to maintain robust prognostic performance even when transcriptomic data are absent. In this work, we propose the Adaptive Multi-modality Knowledge Distillation (AMKD), which enables robust survival prediction using only pathology slides during inference. AMKD introduces (1) a gene-guided pathology knowledge enhancement module that refines multi-modal knowledge from the multi-model teacher, and (2) an adaptive redundancy reduction loss that dynamically balances knowledge transfer based on discrepancies between teacher-student performance during prediction. Evaluated on four TCGA datasets, AMKD achieves state-of-the-art performance (average C-index: 0.669), outperforming both unimodal and multi-modal methods.