Ultrasound Computed Tomography (USCT) has emerged as a cutting-edge imaging modality, offering quantitative acoustic parameter maps to enhance disease diagnosis. Full-waveform Inversion (FWI), a mainstream reconstruction method, enables high-resolution imaging of the speed of sound (SOS) from USCT measurements. However, its strong sensitivity to the initial model and the anatomical distortions caused by cycle-skipping artifacts significantly hinder its application in complex clinical scenarios. In this paper, we propose P \(^{2}\) INR-FWI, a Polar coordinate-based Implicit Neural Representation framework with structural Prior, to achieve unsupervised, subject-specific SOS reconstruction. Departing from conventional Cartesian coordinate-based neural representations, our method introduces a polar coordinate encoding mechanism aligned with the geometry of the USCT ring array, which substantially accelerates convergence and improves reconstruction accuracy. Furthermore, we develop a reflected signal-derived structural prior extraction method to guide the reconstruction process toward clinically critical regions, thereby enabling fine-structure restoration. Experiments conducted on numerical phantom, breast-mimicking phantom, and in vivo data demonstrate that our method outperforms traditional approaches in both reconstruction quality and quantitative metrics, without requiring additional regularization constraints.