Deployable service and delivery robots struggle to navigate multi-floor buildings to reach object goals, as existing systems fail due to single-floor assumptions and requirements for offline, globally consistent maps. Multi-floor environments pose unique challenges including cross-floor transitions and vertical spatial reasoning, especially navigating unknown buildings. Object-Goal Navigation benchmarks like HM3D and MP3D also capture this multi-floor reality, yet current methods lack support for online, floor-aware navigation. To bridge this gap, we propose \textbf{\textit{ASCENT}}, an online framework for Zero-Shot Object-Goal Navigation that enables robots to operate without pre-built maps or retraining on new object categories. It introduces: (1) a \textbf{Multi-Floor Abstraction} module that dynamically constructs hierarchical representations with stair-aware obstacle mapping and cross-floor topology modeling, and (2) a \textbf{Coarse-to-Fine Reasoning} module that combines frontier ranking with LLM-driven contextual analysis for multi-floor navigation decisions. We evaluate on HM3D and MP3D benchmarks, outperforming state-of-the-art zero-shot approaches, and demonstrate real-world deployment on a quadruped robot.