For reference, I just tested this sequence and it gives you a website accessible over the Internet:
- Create a new Windows Azure virtual machine with the Windows Server 2008 R2 SP1 image.
- Add an endpoint on public port 80, private port 80.
- While the endpoint is being created, start setting the server up.
- Remote Desktop in.
- Add the Web Server (IIS) role with default settings.
- Test the connection. You should get a HTTP 200 OK status.
If you want to troubleshoot your server, start checking for errors in the event log. Check also the website bindings in IIS (Port 80, IP Address *).
Also consider the connection issue might be on the client (your) side. For instance, DNS caching. Try connecting from another machine with direct Internet connection (such as another cloud server) or from a service such as isup.me.
Additionally, if all you want is to host websites in IIS, the Web Sites service has a more streamlined experience.