Hi,
I have developed software for over 20 years, and iOS apps for over 3 years.
You can see some of the apps I wrote by searching for "naxomedia" on the App Store. While very different from your project, they illustrate the use of many technologies, including push notifications.
I'm not quite sure what you mean by "auto run". Apple imposes quite a few restrictions on what apps can or cannot do, so depending on what you have in mind, it may or may not be possible. If you want the app to come up spontaneously on the device, that's definitely not possible. It's however possible for the app to display a notification on the device, and the user can then launch the app. If you are thinking about doing stuff in the background, then some things are allowed by Apple for some times of apps, while others are not.
Given the lack of detail of your project description, it is quite difficult to give an accurate cost and time, so those I quoted are of course subject to variation once I get more details about what you want to achieve.
Let me know if you can clarify this or have any questions.
Jacques.