Information diffusion prediction (IDP) is a pivotal task for understanding the dynamics of information propagation within social networks. Conventional models typically adhere to a fixed learning-based paradigm, where the trained prediction model remains static during the inference phase. This paradigm presupposes that the data is independent and identically distributed, an assumption that may not hold true due to the inherently open nature of social media and the uncertainty and variability in user behavior. In this paper, we address the novel problem of out-of-distribution (OOD) shifts within IDP tasks and propose a new test-time training-based model for multi-scale IDP tasks, named Ghidorah. Our approach focuses on adapting a subset of model parameters to accommodate the unique characteristics of test samples through self-supervised learning (SSL) tasks. Ghidorah comprises three components: the macroscopic prediction branch, the microscopic prediction branch, and the auxiliary SSL branch. The auxiliary SSL task employs a masked autoencoder-based loss to fine-tune the model for specific test samples prior to prediction. Furthermore, Ghidorah integrates invariant learning to capture robust representations while mitigating spurious correlations. To our knowledge, Ghidorah is the first work to introduce a test-time training framework specifically designed to address the critical yet often overlooked OOD challenges in IDP. Experimental results across several benchmark datasets validate the superiority of our approach.