Addressing missing modalities is a critical challenge in multimodal brain tumor segmentation. Most existing approaches merely handle modality-incomplete inputs during inference, assuming a full set of modalities for all training samples. However, this unrealistic assumption limits the usage of abundant modality-incomplete data commonly observed in clinical practice. In this paper, we explore a more practical task of tackling missing modalities during both training and inference. We propose a universal model featuring robust modality reconstruction and prompt-guided modality adaptation. Our mask-reconstruction pre-training enables robust modality-invariant representation learning, during which we design a novel distribution approximation method that supervises the reconstruction of absent modalities without requiring full-modal training data. Afterwards, when adapting our model to the segmentation task, we introduce the complete-then-distill (CTD) paradigm, which first estimates missing modalities in training samples from the available ones, and then distills the knowledge from the reconstructed full-modal representations to enhance learning from modality-incomplete data. Moreover, we propose prompt-guided modality adaptation to personalize a subset of model parameters during CTD, enabling the model to adapt to each distinct modality input scenario by using prompts with rich visual-textual information. Extensive experiments on two brain tumor segmentation benchmarks show our method consistently surpasses previous state-of-the-art approaches under dual-stage missing modality settings across various missing ratios.