Urban Air Mobility (UAM) promises a new dimension to decongested, safe, and fast travel in urban and suburban hubs. These UAM aircraft are conceived to operate from small airports called vertiports each comprising multiple take-off/landing and battery-recharging spots. Since they might be situated in dense urban areas and need to handle many aircraft landings and take-offs each hour, managing this schedule in real-time becomes challenging for a traditional air-traffic controller but instead calls for an automated solution. This paper provides a novel approach to this problem of Urban Air Mobility - Vertiport Schedule Management (UAM-VSM), which leverages graph reinforcement learning to generate decision-support policies. Here the designated physical spots within the vertiport's airspace and the vehicles being managed are represented as two separate graphs, with feature extraction performed through a graph convolutional network (GCN). Extracted features are passed onto perceptron layers to decide actions such as continue to hover or cruise, continue idling or take-off, or land on an allocated vertiport spot. Performance is measured based on delays, safety (no. of collisions) and battery consumption. Through realistic simulations in AirSim applied to scaled down multi-rotor vehicles, our results demonstrate the suitability of using graph reinforcement learning to solve the UAM-VSM problem and its superiority to basic reinforcement learning (with graph embeddings) or random choice baselines.