In the realm of modern service-oriented architecture, ensuring Quality of Service (QoS) is of paramount importance. The ability to predict QoS values in advance empowers users to make informed decisions, ensuring that the chosen service aligns with their expectations. This harmonizes seamlessly with the core objective of service recommendation, which is to adeptly steer users towards services tailored to their distinct requirements and preferences. However, achieving accurate and real-time QoS predictions in the presence of various issues and anomalies, including outliers, data sparsity, grey sheep instances, and cold start scenarios, remains a challenge. Current state-of-the-art methods often fall short when addressing these issues simultaneously, resulting in performance degradation. In response, in this paper, we introduce an anomaly-resilient real-time QoS prediction framework (called ARRQP). Our primary contributions encompass proposing an innovative approach to QoS prediction aimed at enhancing prediction accuracy, with a specific emphasis on improving resilience to anomalies in the data. ARRQP utilizes the power of graph convolution techniques, a powerful tool in graph-based machine learning, to capture intricate relationships and dependencies among users and services. By leveraging graph convolution, our framework enhances its ability to model and seize complex relationships within the data, even when the data is limited or sparse. ARRQP integrates both contextual information and collaborative insights, enabling a comprehensive understanding of user-service interactions. By utilizing robust loss functions, this approach effectively reduces the impact of outliers during the training of the predictive model. Additionally, we introduce a method for detecting grey sheep users or services that is resilient to sparsity. These grey sheep instances are subsequently treated separately for QoS prediction. Furthermore, we address the cold start problem as a distinct challenge by emphasizing contextual features over collaborative features. This approach allows us to effectively handle situations where newly introduced users or services lack historical data. Experimental results on the publicly available benchmark WS-DREAM dataset demonstrate the framework's effectiveness in achieving accurate and timely QoS predictions, even in scenarios where anomalies abound.