We use a data-driven simulation framework to constrain the origin of individual ultra-high-energy cosmic rays (UHECRs) through 3D modeling and likelihood-free inference. We highlight how this method can identify potential source locations based on propagation effects through Galactic and extra-Galactic magnetic fields and emphasize the role of systematic uncertainties. Our approach relies on CRPropa3 to simulate propagation and uses Approximate Bayesian Computation to derive constraints on key source parameters including position, distance, energy, and the properties of intervening magnetic fields. As a case study, we apply our methodology to the Amaterasu particle, the second-highest energy cosmic ray ever detected. We present our results, demonstrating the impact of the systematic uncertainty on the energy and that of different assumptions for the arrival mass of the Amaterasu particle.