Olde

The internet is complex - intentionally if you draw out the design on an A3 piece of paper. Before thinking about keeping the internet up and running you would need to have power - if the power is running then you would need to be able to effectively clone the internet root server farm. This is where the address ranges are managed. The clone would need to be both hard and soft components + all data and metadata.
You would then need to ensure you have control and power to all ISP's and content provider server farms and networks to ensure data and data transport are running. In a sense it is impossible to keep the whole internet running if society collapsed. If you want to keep this and some selective sites running that is a different matter and could be done but the cost is amazing and would involve control over some infrastructure in the skies and on the ground that is very difficult to get.