Numerical simulations of simplex pressure-swirl atomizers can aid their design process toward better atomization. This work aims at studying the two-phase flow at both the internal geometry and the first millimeters of the external domain of such atomizers, where primary breakup takes place. In particular, the atomizer under study has been used in the CORIA Rouen Spray Burner (CRSB) test rig, which aims at studying lean premixed turbulent combustion. Ultimately, our goal is to complete the spray characterization in the vicinity of the injector. Such data will potentially enforce the validity of numerical simulations of nonreacting and reacting flow for this burner. Injection characteristics are analyzed through an interface capturing method within a detailed numerical simulations framework. The importance of the internal flow simulation on the final result is demonstrated in the manuscript, but it requires accurately measuring the injector internal geometry. In the present investigation, an experimental methodology combining different techniques is applied to this end, obtaining and parameterizing the actual geometry of the internal ducts within the atomizer. The numerical workflow is divided in two simulations to separately study the internal flow formation and the external spray development. This division is proposed given the difficulty to mesh the whole computational domain handling all the present length scales while still preserving the required accuracy. Several mesh refinements are studied for each simulation, also analyzing the coupling between the related internal and external simulations. The methodology is validated against experimental data for two CRSB operating conditions. The investigation then proves it is possible to couple the internal and external flow in order to describe the actual air core formation, liquid film behavior and breakup mechanism of these atomizers, extracting relevant atomization outputs in the near-field region where experimental data are scarce.