What are the options to display a shapefile with over 1000 features in ArcGIS Online?

Answer

The following are the two available options for displaying a shapefile with over 1,000 features in ArcGIS Online:

Via ArcGIS Online

By design, a shapefile containing more than 1,000 features cannot be added to an ArcGIS Online web map. However, users can publish a hosted feature layer instead of adding the file to the web map. For more details, refer to the following ArcGIS Online help page: Publish features.
To change the default maximum records displayed once a service has been published, navigate to the service REST endpoint. Refer to the following article for more details: HowTo: Update the maximum record count for feature services in ArcGIS Online.

Via ArcMap

Increase the number of features in the Parameters option before publishing the map features to ArcGIS Online.

1. In ArcMap, open a map with the shapefile.
2. Click File > Sign In, and log into ArcGIS Online.
3. Click File > Share As > Service.
4. Select 'Publish a service', and click Next.
5. Select My Hosted Services in the 'Choose a connection' drop-down list, and enter the desired name for the service in the Service name field. Click Continue.
6. In the Service Editor, click Capabilities in the left pane, and check the Feature Access option to expose the service as a feature service.
7. Click Parameters, and enter a larger value for the 'Maximum number of records returned by the server' (the default is 1,000). Set the number of records to the desired value.

8. Click Item Description, and enter a summary and tags for the service.
9. Click Sharing, and select which users to share the service with.
10. In the Service Editor window, click Analyze to examine the map to be published as a service, and click Preview to view how the service appears on the web.
11. Click Publish.