Information diffusion prediction (IDP) is a pivotal task for understanding how information propagates among users. Most existing methods commonly adhere to a conventional training-test paradigm, where models are pretrained on training data and then directly applied to test samples. However, the success of this paradigm hinges on the assumption that the data are independently and identically distributed, which often fails in practical social networks due to the inherent uncertainty and variability of user behavior. In the paper, we address the novel challenge of distribution shifts within IDP tasks and propose a robust test-time training (TTT)-based framework for multi-scale diffusion prediction, named T3MAL. The core idea is to flexibly adapt a trained model to accommodate the distribution of each test instance before making predictions via a self-supervised auxiliary task. Specifically, T3MAL introduces a BYOL-inspired self-supervised auxiliary network that shares a common feature extraction backbone with the primary diffusion prediction network to guide instance-specific adaptation during testing. Furthermore, T3MAL enables fast and accurate test-time adaptation by incorporating a novel meta-auxiliary learning scheme and a lightweight adaptor, which together provide better weight initialization for TTT and mitigate catastrophic forgetting. Extensive experiments on three public datasets demonstrate that T3MAL outperforms various state-of-the-art methods.