I need the OWA web part to automatically login the current user. How to do so using C#
This really isn't something you can develop a solution for unless you are looking to build your own Single Sign-On solution (not recommended). Take a look at this other thread that outlines two options (Kerberos delegation or ADFS).
Please check out this post and my answer.
Sharepoint 2010 / Exchange 2010 Single Sign On issue
If the user accessing SharePoint is the same account for the OWA access it is all a matter of configuration for the client and the servers (Exchange and SharePoint).
If you have the need to do it via code this will be a pretty big effort you will have to take with deep knowledge of kerberos and ad in general.