You are right, you need to build a web app which automatically searches Twitter and then creates a push notification payload to deliver to your users. In addition, you may want your app to also accept requests from users to retrieve the actual tweets. This is because you can't just load up the tweet content into a notification payload, as their is a rather small size limit.
Since Urban Airship works via HTTP, you can use whatever the heck language you want. Just go with what you know, there is no preferred way to do it.
I will say this though - there are lots and lots of tweets added every second. Scalability is probably going to play a pretty big factor in whatever you are trying to do. Which is probably why a year later there aren't very many Twitter apps that support Push.