Hi Eninna,
A map package is used to share/ use in ArcGIS Desktop. To open in ArcGIS Online, you need to share the map as a service.
1. Create an ArcGIS Online organizational account, if not created. You could also try using a 60 days trial ArcGIS Online account with 200 credits.
ArcGIS Platform | Free Trial
2. Sign-in to ArcGIS Online from ArcMap.
3. Share the map as a service. Choose your hosted organizational account.Select feature access capability and deselect tiled option. Then analyze and publish. The service will be added to the hosted services.
Also I would suggest you to use the web mercator projection system for the data frame properties using appropriate transformation method.
Think Location