You should be able to use featureTableForLayerName to retrieve non-spatial tables from the geodatabase. These would be returned just like the spatial tables as AGSGDBFeatureTable objects. There is no difference in querying spatial or non-spatial tables. You use the same class and the same methods, the only difference being that the results (AGSGDBFeature objects) won't have a geometry for non-spatial tables.
Where did you create the geodatabase from? ArcMap or from a feature service?