Isolated cold-start node classification on multimodal graphs is challenging because such nodes have no edges and often have missing modalities (e.g., absent text or image features). Existing methods address structural isolation by degrading graph learning models to MLPs for isolated cold-start inference, using a teacher model (with graph access) to guide the MLP. However, this results in limited model capacity in the student, which is further challenged when modalities are missing. In this paper, we propose Neighbor-to-Self Graph Transformer (NTSFormer), a unified Graph Transformer framework that jointly tackles the isolation and missing-modality issues via a self-teaching paradigm. Specifically, NTSFormer uses a cold-start attention mask to simultaneously make two predictions for each node: a "student" prediction based only on self-information (i.e., the node's own features), and a "teacher" prediction incorporating both self and neighbor information. This enables the model to supervise itself without degrading to an MLP, thereby fully leveraging the Transformer’s capacity to handle missing modalities. To handle diverse graph information and missing modalities, NTSFormer performs a one-time multimodal graph pre-computation that converts structural and feature data into token sequences, which are then processed by Mixture-of-Experts (MoE) Input Projection and Transformer layers for effective fusion. Experimental results on public datasets show that NTSFormer achieves superior performance on multimodal isolated cold-start node classification tasks. Our code is provided.