The ability to handle outliers is essential for performing the perspective- n-point (P nP) approach in practical applications, but conventional RANSAC+P3P or P4P methods have high time complexities. We propose a fast P nP solution named R1PP nP to handle outliers by utilizing a soft re-weighting mechanism and the 1-point RANSAC scheme. We first present a P nP algorithm, which serves as the core of R1PP nP, for solving the P nP problem in outlier-free situations. The core algorithm is an optimal process minimizing an objective function conducted with a random control point. Then, to reduce the impact of outliers, we propose a reprojection error-based re-weighting method and integrate it into the core algorithm. Finally, we employ the 1-point RANSAC scheme to try different control points. Experiments with synthetic and real-world data demonstrate that R1PP nP is faster than RANSAC+P3P or P4P methods especially when the percentage of outliers is large, and is accurate. Besides, comparisons with outlier-free synthetic data show that R1PP nP is among the most accurate and fast P nP solutions, which usually serve as the final refinement step of RANSAC+P3P or P4P. Compared with REPP nP, which is the state-of-the-art P nP algorithm with an explicit outliers-handling mechanism, R1PP nP is slower but does not suffer from the percentage of outliers limitation as REPP nP.