1. Internet connection errors that appear when opening the app for the first time
My proposal is to handle these errors with error screens (most of the apps that I benchmarked are doing the same). The problem is that in this case there isn't anything cached in your app and you cannot provide content to the user. Therefore you need to ask them to quite clearly to do something about this.
However you should be careful to load some content (top bar or maybe basic navigation) just enough to communicate that there isn't something wrong with the app and put the users in the correct context.
2. Internet connection errors that appear while you are using the app and can affect requests to the server
My proposal is to handle these error in context with a bottom notification (gmail app is doing the same). The reasoning behind this is that you don't interrupt the user experience (users can still interact with the cached content) but in the same time you are informing them that something is wrong. Then if they end up submitting something or trying to navigate to a page where a new request is made you should go for the 1st solution.