Zero-shot reasoning on text-rich networks (TRNs) remains a challenging frontier, as models must integrate textual semantics with relational structure without task-specific supervision. While graph neural networks rely on fixed label spaces and supervised objectives, recent large language model (LLM)-based approaches often overlook graph context or depend on distillation from larger models, limiting generalisation. We propose TRN-R1-Zero, a post-training framework for TRN reasoning trained solely via reinforcement learning. TRN-R1-Zero directly optimises base LLMs using a Neighbour-aware Group Relative Policy Optimisation objective that dynamically adjusts rewards based on a novel margin gain metric for the informativeness of neighbouring signals, effectively guiding the model toward relational reasoning. Unlike prior methods, TRN-R1-Zero requires no supervised fine-tuning or chain-of-thought data generated from large reasoning models. Extensive experiments across citation, hyperlink, social and co-purchase TRN benchmarks demonstrate the superiority and robustness of TRN-R1-Zero. Beyond cross-domain transfer, TRN-R1-Zero, trained solely on node-level tasks, further generalises to edge- and graph-level tasks in a zero-shot manner. The codebase is open-source at [https://github.com/superallen13/TRN-R1-Zero](https://github.com/superallen13/TRN-R1-Zero).