Aerial-ground interference mitigation is a challenging issue in the cellular-connected unmanned aerial vehicle (UAV) communications. Due to the strong line-of-sight (LoS) air-to-ground (A2G) channels, the UAV may impose/suffer more severe uplink/downlink interference to/from the cellular base stations (BSs) than the ground users. To tackle this challenge, we propose to apply the non-orthogonal multiple access (NOMA) technique to the uplink communication from a UAV to cellular BSs, under spectrum sharing with the existing ground users. However, for our considered system, traditional NOMA with local interference cancellation (IC), termed non-cooperative NOMA, may provide very limited gain compared to the OMA. This is because there are many co-channel BSs due to the LoS A2G channels and thus the UAV's rate performance is severely limited by the BS with the worst channel condition with the UAV. To improve the UAV's achievable rate, a new cooperative NOMA scheme is proposed by exploiting the backhaul links among BSs. Specifically, some BSs with better channel conditions are selected to decode the UAV's signals first, and then forward the decoded signals to their backhaul-connected BSs for IC. To investigate the optimal design of cooperative NOMA, we maximize the weighted sum-rate of the UAV and ground users by jointly optimizing the UAV's rate and power allocations over multiple resource blocks. However, this problem is hard to be solved optimally. To obtain useful insights, we first consider two special cases with egoistic and altruistic transmission strategies of the UAV, respectively, and solve their corresponding problems optimally. Next, we consider the general case and propose an efficient suboptimal solution via the alternating optimization. Numerical results show that the proposed cooperative NOMA yields significant throughput gains than the OMA and the non-cooperative NOMA benchmark.