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Nonlinear Model Predictive Control (NMPC) is a powerful and widely used technique for nonlinear dynamic process control under constraints. In NMPC, the state and control weights of the corresponding state and control costs are commonly selected based on human-expert knowledge, which usually reflects the acceptable stability in practice. Although broadly used, this approach might not be optimal for the execution of a trajectory with the lowest positional error and sufficiently "smooth" changes in the predicted controls. Furthermore, NMPC with an online weight update strategy for fast, agile, and precise unmanned aerial vehicle navigation, has not been studied extensively. To this end, we propose a novel control problem formulation that allows online updates of the state and control weights. As a solution, we present an algorithm that consists of two alternating stages: (i) state and command variable prediction and (ii) weights update. We present a numerical evaluation with a comparison and analysis of different trade-offs for the problem of quadrotor navigation. Our computer simulation results show improvements of up to 70% in the accuracy of the executed trajectory compared to the standard solution of NMPC with fixed weights.