At EuPRAXIA@SPARC_LAB, the unique combination of an advanced high-brightness RF injector and a plasma-based accelerator will drive a new multi-disciplinary user-facility. The facility, that is currently under study at INFN-LNF Laboratories (Frascati, Italy) in synergy with the EuPRAXIA collaboration, will operate the plasma-based accelerator in the external injection configuration. Since in this configuration the stability and reproducibility of the acceleration process in the plasma stage is strongly influenced by the RF-generated electron beam, the main challenge for the RF injector design is related to generating and handling high quality electron beams. In the last decades of R&D activity, the crucial role of high-brightness RF photo-injectors in the fields of radiation generation and advanced acceleration schemes has been largely established, making them effective candidates to drive plasma-based accelerators as pilots for user facilities. An RF injector consisting in a high-brightness S-band photo-injector followed by an advanced X-band linac has been proposed for the EuPRAXIA@SPARC_LAB project. The electron beam dynamics in the photo-injector has been explored by means of simulations, resulting in high-brightness, ultra-short bunches with up to 3 kA peak current at the entrance of the advanced X-band linac booster. The EuPRAXIA@SPARC_LAB high-brightness photo-injector is described here together with performance optimisation and sensitivity studies aiming to actual check the robustness and reliability of the desired working point.