I think your app would have to run in the same application context as mojoPortal to read its cookies.
string userName = Context.User.Identity.Name;
If you add this in user.config:
<add key="EnableDeveloperMenuInAdminMenu" value="true" />
it enables a tool in mojoPortal to see all the server variables under Administration > Advanced Tools > Developer Tools > Server Variables
However depending on configuration what you see there may not be the same as what you will see in your custom app.