Here is the answer I got from Esri Technical Support:
It is my understanding that you want to know if you can publish a service to ArcGIS Online using ArcMap 10.0. I have looked into this and that feature was implemented starting at version 10.1. https://community.esri.com/thread/44828
If you have access to a later version that you can use for your publishing that would be the easiest solution. If you do not there are other ways you can publish a service. For example, you can zip a shapefile and upload this to ArcGIS Online and from there publish this to a feature service. Please see the following documentation and keep in mind that some of these other options have limitations: http://doc.arcgis.com/en/arcgis-online/create-maps/add-layers.htm
