Hi Alexis,
Can you please provide more information:
Are you publishing from ArcMap directly to ArcGIS Online and therefore creating a hosted feature layer? or
Are you publishing from ArcMap to a GIS Server site; then referencing the web service in your web map?
If you are publishing directly to ArcGIS Online and creating hosted feature layers, then the data is always copied to ArcGIS Online and there is no way to automatically update the data - you would have to republish the updated data. I suppose this might be automated with a custom python script.
If you are publishing to a GIS Server site, have you defined data stores (e.g., a valid location for source data for your web services) for the GIS Server site? Registering your data with the GIS Server site will enable you to reference your source data for the web services. This means that as the source data is updated, the web service is updated, and data in the web map will also be updated.
Hope this helps,