Non-linear model predictive control (NMPC) solves structured optimization problems under predetermined constraints. It results in an optimal control series in a multiple-step prediction horizon. However, the NMPC requires considerable computation time, making it difficult to implement on devices with limited resources. We focus on an NMPC-based controller used for autonomous vehicle racing. It is a typical representative of quickly evolving cyber-physical systems. In the single-vehicle racing mode, we propose a triggering method to enable the execution of long-horizon NMPC, which is desirable for achieving a better lap time. For the head-to-head racing mode, in order to react rapidly to the evolving surroundings, we propose a short-horizon NMPC-based control strategy with safe overtaking capability. These control strategies can be implemented within a limited time budget.