Try looking at WaTiN.
It's a framework we use to perform tests on our webapplication, mimicing user interaction. We also use to to communicate with Mailinator to check dummy emails.
I'm pretty sure WaTiN can be of help here, but the downside is that it will actually open a specific browser, so if you can live with that, it's the way to go.
You could also make webrequests and provide the correct post data to perform a login, it's not that difficult really. Afterwards, just parse the content of the response to check if there's email.
You used to be able to use the System.Net.mail namespace and mailclients to connect to the hotmail servers directly, but i'm not sure if hotmail still allows this.