The core-collapse supernova (CCSN) is considered one of the most energetic astrophysical events in the universe. The early and prompt detection of neutrinos before (pre-SN) and during the supernova (SN) burst presents a unique opportunity for multi-messenger observations of CCSN events. In this study, we describe the monitoring concept and present the sensitivity of the system to pre-SN and SN neutrinos at the Jiangmen Underground Neutrino Observatory (JUNO), a 20 kton liquid scintillator detector currently under construction in South China. The real-time monitoring system is designed to ensure both prompt alert speed and comprehensive coverage of progenitor stars. It incorporates prompt monitors on the electronic board as well as online monitors at the data acquisition stage. Assuming a false alert rate of 1 per year, this monitoring system exhibits sensitivity to pre-SN neutrinos up to a distance of approximately 1.6 (0.9) kiloparsecs and SN neutrinos up to about 370 (360) kiloparsecs for a progenitor mass of 30 solar masses, considering both normal and inverted mass ordering scenarios. The pointing ability of the CCSN is evaluated by analyzing the accumulated event anisotropy of inverse beta decay interactions from pre-SN or SN neutrinos. This, along with the early alert, can play a crucial role in facilitating follow-up multi-messenger observations of the next galactic or nearby extragalactic CCSN.