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 plasmabased 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 photoinjector 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.