Hi Adriana,
Publish a custom network dataset to ArcGIS Online isn't supported. Publish custom network dataset and use for routing service is only supported through ArcGIS Enterprise.
Once the services are published and hosted on ArcGIS Enterprise, the route service in ArcGIS Online organization can be configured to use the route service from ArcGIS Enterprise, if ArcGIS Online org can access your service from your ArcGIS Enterprise. The only out of box client app that can consume such service is the directions widget. So the functionality is really limited. The best way to use routing service with custom network dataset is directly through ArcGIS Enterprise.
