Do not drop the multiplayer idea! It's way cooler with multiplayer. :)
The greatest advantage of socket.io is the fallback. You can set it to start with any protocol (like WebSockets) and if the client does not support it, then it tries the others. It is really great, since you can use WebSockets (which will slowly but surely dominate web apps) and still work with browsers which do not support it, like IE or Opera or Safari. I don't know whether there is any other library with this advantage.
I don't know any library for real-time connections for C# (I'm not a C# developer), but it is unlikely that there are none (look at this question). Also note that real-time connections require a bit different server architecture then normal HTTP requests, so probably you will need additional server for handling them.
Also I think that neither nginx nor Apache handle WebSockets (without some hard core tricks) at the moment (but Node JS does!). I'm not sure though.
@MetalFrog According to wiki: "it (...) has been disabled in Opera 11 until the new version of the protocol is implemented." which actually makes it useless. I do not know the current situation though.
opera:config#Enable%20WebSockets will take care of that.
@MetalFrog You are not a normal user (being a user of stackoverflow is a proof :)) 99% of users are lazy. And most of them would like to have everything ready and working out of the box, especially via browser. After telling them: "install this, install that, add this plugin, configure that, etc." you automatically lose many (sometimes most) of them and make your site less attractive. That's at least what I've seen in projects I worked on.