Efficient and accurate brain ventricle segmentation from clinical CT scans is critical for emergency surgeries like ventriculostomy. With the challenges in poor soft tissue contrast and a scarcity of well-annotated databases for clinical brain CTs, we introduce a novel uncertainty-aware ventricle segmentation technique without the need of CT segmentation ground truths by leveraging diffusion-model-based domain adaptation. Specifically, our method employs the diffusion Schr\"odinger Bridge and an attention recurrent residual U-Net to capitalize on unpaired CT and MRI scans to derive automatic CT segmentation from those of the MRIs, which are more accessible. Importantly, we propose an end-to-end, joint training framework of image translation and segmentation tasks, and demonstrate its benefit over training individual tasks separately. By comparing the proposed method against similar setups using two different GAN models for domain adaptation (CycleGAN and CUT), we also reveal the advantage of diffusion models towards improved segmentation and image translation quality. With a Dice score of 0.78$\pm$0.27, our proposed method outperformed the compared methods, including SynSeg-Net, while providing intuitive uncertainty measures to further facilitate quality control of the automatic segmentation outcomes.