As a pioneering km$^{3}$-scale neutrino detector, the IceCube Observatory has established the feasibility of high-energy neutrino astrophysics by confirming the existence of astrophysical neutrinos, and has provided evidence for several potential extragalactic sources, such as TXS 0506$+$056, NGC 1068, and NGC 4151. To advance the field further, neutrino telescopes with a larger volume and higher precision are essential. To this end, we proposed the NEutrino Observatory in the Nanhai (NEON), a detector to be deployed in the South China Sea with a physical volume of about 10 km$^{3}$, which is currently under development. Given the anticipated high performance of NEON, we considered non-jetted active galactic nuclei (AGNs) as one of the most promising targets. By adopting specific models including the magnetically-powered corona model and the radiatively inefficient accretion flow (RIAF) model, we evaluated the detectability of such extragalactic sources with NEON and calculated the expected all-flavor neutrino event number after oscillation for 10 years of observation. The results demonstrate that NEON should be able to detect extragalactic sources at distances of up to 300 Mpc. Among these, a source with the X-ray luminosity and distance of NGC 1068 represents a prime candidate, especially in the several-to-hundred TeV energy range. Moreover, we have conducted preliminary tests on the key hardware components for the NEON experiment. The reliability of the photomultiplier tubes (PMTs), front-end and readout electronics, as well as mechanical structures has been verified by these trials.