Black hole accretion in active galactic nuclei (AGN) is coupled to the evolution of their host galaxies. Outflowing winds in AGN can play an important role in this evolution through the resulting feedback mechanism. Multi-wavelength spectroscopy is key for probing the intertwined physics of inflows and outflows in AGN. However, with the current spectrometers, crucial properties of the ionized outflows are poorly understood, such as their coupling to accretion rate, their launching mechanism, and their kinetic power. We discuss the need for simultaneous X-ray and UV high-resolution spectroscopy for tackling outstanding questions on these outflows in AGN. The instrumental requirements for achieving the scientific objectives are addressed. We demonstrate that these requirements would be facilitated by the proposed Arcus Probe mission concept. The multi-wavelength spectroscopy and timing by Arcus would enable us to establish the kinematics and ionization structure of the entire ionized outflow, extending from the vicinity of the accretion disk to the outskirts of the host galaxy. Arcus would provide key diagnostics on the origin, driving mechanism, and energetics of the outflows, which are useful benchmarks for testing various theoretical models of outflows and understanding their impact in AGN.