Ultrasound computed tomography (UCT) via full waveform inversion (FWI) enables high-resolution quantitative imaging for tissue characterization and disease diagnosis. However, UCT suffers from large computational burden and severe convergence issues due to highly nonlinear optimization. Deep learning can accelerate UCT reconstruction, but supervised training requires large-scale labeled datasets difficult to obtain in vivo. To address these limitations, we propose SDA-UCT, a two-stage self-supervised domain-adaptive framework for rapid and accurate UCT imaging of musculoskeletal tissues. SDA-UCT employs an attention-enhanced network (AttUCT) pre-trained on simulation datasets and transfers to in-vivo data via physics-informed self-supervised learning, effectively bridging the simulation-to-real domain gap. A Low-Rank Adaptation (LoRA) mechanism is integrated to enable efficient adaptation across diverse clinical scenarios. Results showed that AttUCT achieved high-quality SOS reconstruction for simulated human forearm with a PSNR of 29.23 dB and SSIM of 0.928, outperforming conventional FWI and existing deep learning methods. Validated on in-vivo data, SDA-UCT successfully reconstructed SOS images revealing complex anatomical structures (skin, fat, muscle, tendon, bone and bone marrow) for human forearm, in high concordance with MRI references. The LoRA mechanism adjusting only 3% of parameters achieved comparable performance to full fine-tuning. The rapid reconstruction (5 ms per frame) enables real-time 3D visualization, achieving five-orders-of-magnitude improvement over traditional FWI. This work represents the first self-supervised domain-adaptive deep learning for rapid, high-resolution in-vivo UCT imaging, showing potential for musculoskeletal disease diagnosis.