The RadMap control provides an out-of-the-box logic to render a map using mathematical algorithms and images fetched by a third-party web-services (e.g., OpenStreetMap).
Yes, this web service needs internet connection, so that the images could be requested. If there is a possible approach for such service to be created in a local environment (e.g., Intranet scenario), this is a matter that is not somehow related to the RadMap, but to a custom Map Tile Image provider. Such matter is discussed in this OpenStreetMap thread
About showing only a city map. You cannot somehow restrict the Map to not show the entire world map, but only one city. Note that when the proper center setting and zoom level are set according to the desired city to be shown initially, only the images corresponding to the viewport size will be requested, therefore additional images are not being downloaded to the client.
Based on that, I can suggest just configuring the Map to be initially centered on the desired city.
Check out the Telerik Platform - the only platform that combines a rich set of UI tools with powerful cloud services to develop web, hybrid and native mobile apps.