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Knowledge discovered from telecom data can facilitate proactive understanding of network dynamics and user behaviors, which in turn empowers service providers to optimize cellular traffic scheduling and resource allocation. Nevertheless, the telecom industry still heavily relies on manual expert intervention. Existing studies have been focused on exhaustively explore the spatial-temporal correlations. However, they often overlook the underlying characteristics of cellular traffic, which are shaped by the sporadic and bursty nature of telecom services. Additionally, concept drift creates substantial obstacles to maintaining satisfactory accuracy in continuous cellular forecasting tasks. To resolve these problems, we put forward an online cellular traffic prediction method grounded in Multi-Grained Spatial-Temporal feature Complementarity (MGSTC). The proposed method is devised to achieve high-precision predictions in practical continuous forecasting scenarios. Concretely, MGSTC segments historical data into chunks and employs the coarse-grained temporal attention to offer a trend reference for the prediction horizon. Subsequently, fine-grained spatial attention is utilized to capture detailed correlations among network elements, which enables localized refinement of the established trend. The complementarity of these multi-grained spatial-temporal features facilitates the efficient transmission of valuable information. To accommodate continuous forecasting needs, we implement an online learning strategy that can detect concept drift in real-time and promptly switch to the appropriate parameter update stage. Experiments carried out on four real-world datasets demonstrate that MGSTC outperforms eleven state-of-the-art baselines consistently.