Different from public 4G/5G networks that are dominated by downlink traffic, emerging 5G non-public networks (NPNs) need to support significant uplink traffic to enable emerging applications such as industrial Internet of things (IIoT). The uplink-and-downlink spectrum sharing is becoming a viable solution to enhance the uplink throughput of NPNs, which allows the NPNs to perform the uplink transmission over the time-frequency resources configured for downlink transmission in coexisting public networks. To deal with the severe interference from the downlink public base station (BS) transmitter to the coexisting uplink non-public BS receiver, we propose an adaptive asymmetric successive interference cancellation (SIC) approach, in which the non-public BS receiver is enabled to have the capability of decoding the downlink signals transmitted from the public BS and successively cancelling them for interference mitigation. In particular, this paper studies a basic uplink-and-downlink spectrum sharing scenario when an uplink non-public BS and a downlink public BS coexist in the same area, each communicating with multiple users via orthogonal frequency-division multiple access (OFDMA). Under this setup, we aim to maximize the common uplink throughput of all non-public users, under the condition that the downlink throughput of each public user is above a certain threshold. The decision variables include the subcarrier allocation and user scheduling for both non-public (uplink) and public (downlink) BSs, the decoding mode of the non-public BS over each subcarrier (i.e., with or without SIC), as well as the rate and power control over subcarriers. Numerical results show that the proposed adaptive asymmetric SIC design significantly improves the common uplink throughput as compared to benchmark schemes without such design.