Non-orthogonal multiple access (NOMA) and energy-harvesting (EH) relay have been envisioned as promising technologies in Narrowband Internet of Things (NB-IoT) networks to efficiently improve the spectral-energy efficiency of networks and the massive connectivity of devices. However, the successive interference cancellation (SIC) ordering of NOMA has become a bottleneck affecting the performance of uplink transmission in NB-IoT networks. Also, while using NOMA, the data rate based fairness across NB-IoT devices cannot be ensured due to the difference between individual channels. To address these issues, this paper investigates the integration of NOMA and EH relay into the NB-IoT network. Correspondingly, we aim to maximize the data rate based proportional fairness across all NB-IoT devices by jointly optimizing the resource allocation and SIC ordering. We first prove the NP-hardness of this joint optimization problem and then tackle this mathematically intractable problem by decomposing it into resource allocation and SIC ordering problems. To solve the resource allocation problem, we exploit the convexity of the problem through transformation and reparameterization and then derive the optimal solution based on its Lagrangian function. We also propose a low-complexity SIC ordering algorithm by leveraging tabu search to obtain the sub-optimal SIC ordering. Simulation results validate the effectiveness of the proposed algorithm and the superiority of NOMA compared to frequency division multiple access (FDMA).