I have a shapefile on ArcGIS for desktop that contains a few thousand points. I also recently published this shapefile to a new map on ArcGIS online. After working with the map for a while, I realized that some of the points are currently gone now, and there are some points that don't exist that I need to add. Can I somehow edit the shapefile on the desktop and then update the published map online? Or is the only way to change them both to individually edit each of them?
To change your ArcGIS Online (AGOL) service you will need to change your desktop .shp then upload again to AGOL. Making changes to the .shp with ArcGIS for Desktop does not apply changes directly to a service unless you publish the service as a feature service to your ArcGIS for Server and add it as web service in AGOL.
You can set up a feature class in geodatabase in ArcSDE that keeps the two in sync. When you update the feature class in ArcMap, it updates on the published service once you refresh the map service.
Here's an ESRI tutorial on how to get a map service up and running: http://server.arcgis.com/en/server/latest/get-started/windows/tutorial-publishing-a-map-service.htm
And see the "Publish from ArcMap" section of this tutorial: https://doc.arcgis.com/en/arcgis-online/share-maps/publish-features.htm