When publishing a service from ArcMap, go to "Capabilities" and enable "Feature Access" and then select the following under "Operations allowed": Create, Delete, Query, Sync, Update. Your feature should be published
with the correct settings.
If the layer has been published, go to "My Content:" and select the layer (view item details). At the top of the screen will be a series of options that include open, share, edit, etc. Select "edit". Near the bottom is a section for Properties. Check the selections to enable editing and to enable sync and save your changes.
It is also possible to edit the json file and update the section "capabilities" to "Create,Delete,Query,Update,Editing,Sync". I suggest this only if you cannot republish the layer with the sync setting enabled, only after backing up your data and if you are comfortable using the REST API. To edit the json, you will need to insert "admin" in the url: .../arcgis/rest/services/... becomes .../arcgis/rest/admin/services/... (assuming you are using ArcGIS Online). You will need sufficient privileges to do this action. A link "Update Definition" will be available at the bottom of the page.