Büchi Automata on infinite words present many interesting problems and are used frequently in program verification and model checking. A lot of these problems on Büchi automata are computationally hard, raising the question if a learning-based data-driven analysis might be more efficient than using traditional algorithms. Since Büchi automata can be represented by graphs, graph neural networks are a natural choice for such a learning-based analysis. In this paper, we demonstrate how graph neural networks can be used to reliably predict basic properties of Büchi automata when trained on automatically generated random automata datasets.