Here is the trick I use:

A SharePoint user  is actually based on network user (active directory).
When a user is connecting to SharePoint, it connects based on 'Integrated
Windows Authentication'.
You set up your custom web  application to run based on same kind of
authentication (you do the settings from Control Panel -> Admin Tools ->
IIS admin); when the page  Viewer web part  try to open and show on
SharePoint your application  page it will be asked for windows credential
for authentication, your custom web application will be 'hit' with the
same credentials  as SharePoint page. At the start page of your
application (maybe at page_load) just use System.Environment.UserName to
retrieve the user name.