I'm looking for an iOS app to deliver basic text push notifications to my phone (e.g. website downtime alerts or things like that) using its API programmatically. Any recommendations or similar work you know of?
I'm not sure I understood your question. Push notifications have to be originated outside the device. If you want an app running on your device and sending notifications to the device itself, it will be just a regular local notification. If you want to receive notifications from events that will originated outside the device, like the case you mentioned, website downtime alert), some application running running outside the device can use a service like Urban Airship, Amazon AWS, Parse.com and some others. These providers have an API that your app, running on a server somewhere, can use to send push notifications to your iOS device. You would need an app on your device the subscribes to these messages and the app could be launched by the remote notification when the user taps on it. If you are using one of these push notification service providers, your iOS app would have to be linked with their SDK. In iOS, each app that can receive remote push notifications is a topic and the originator of the push notification has to address the notification to a topic/device combination.
This page has a list of several push notification providers: http://iphonedevsdk.com/forum/promotion-techniques/115219-my...
And several of them have tutorials explaining how to use thier service.