Replacement RSS reader for Mountain Lion, integrated with Notification Center?
Firstly, in order to integrate into Notification Centre, an app needs to have been purchased from the App Store.  Only App Store apps can use it (same with most iCloud features).  So your search needs to start there.

2 very popular RSS readers are Reeder and NewsNetWire.  Neither have been update to use Notification Centre yet so far as I know, but for sure some of them will be.

It's likely that if you can find an RSS reader that used to use Growl as it's notification method, then it will be modified in time to use Notification Centre instead, but there are so many RSS readers (particularly on iOS, perhaps less so on the Mac App Store) that I can't name a particular app right now.

I quick search for "rss notification" yields no results, but "rss growl" gives 3 hits (Note that for those who are not aware, Growl is in effect a third party tool that has done what Notification Centre will do for years already, and loads of apps already support it):

http://itunes.apple.com/gb/app/cappuccino/id492594411?mt=12

http://itunes.apple.com/gb/app/monotony/id533978766?mt=12

http://itunes.apple.com/gb/app/feeds/id458374415?mt=12

The fact that these 3 already support some form of notifications may be indicative they they will be amogst the first to support OS X 10.8 Notification Centre.

However there is another way:

For those apps (not just RSS) that use Growl, there is talk that Growl will morph into a sort of bridge between the Notification Centre and non MAS apps, because it is on the MAS itself, and as such can install notifications into the tray, and it can also poll other apps for Growl notifications and simply pass them on without the originating app needing to know about Notification Centre formally.  See this blog post for info.

It's not happened yet tho, although there is already an app to bridge between Growl and Notification Centre called HISS (See here http://9to5mac.com/2012/07/25/put-growl-in-your-notification-center-with-hiss/) which will punt any grown notification into NC instead, although it's very much Beta, and frankly will be superceded as soon as Growl get their own act together, but at least it is available now.

As such you may be able to roll your own temporary bodge using a combination of an RSS reader as suggested above that supports Growl, installing and using Growl, installing Hiss to pass Growl notifications into the Notification Centre...