Hi, John:
I don't know of any RSS reader that'll push you new posts.
However, may I recommend the app "Boxcar," which is an app designed to push you notifications (email, text, Facebook, Twitter, etc). You have to set it up, but one of the things you can set Boxcar up to do is push you a notification that a particular RSS feed has been updated. You'll probably still have to go to that web site, but at least you'll know there's updates.
I don't use Boxcar for RSS feeds (I follow too many), but it works brilliantly for email, text messages and Facebook notices for me. There are a lot of things you can set Boxcar up for within the app - you'll just have to set them up.
The app is, I believe, $2.99 USD in the app store, plus there is a one-time $.99 in-app purchase to get rid of advertisements. Seems a bit pricey, but I feel the benefits are worth it. Of course, and as always, YMMV.
Marilyn