YES getting a 1GB NIC is what you should use to connect the server to your 1GB Switch / Router, that 1GB NIC is for anyone INSIDE your network to run "1GB/1GB Download-Upload " but NOT the Internet.
First off, only in very RARE places (like backbone server farms) actually have a "1GB/1GB Download-Upload" Internet connection, due to the VERY HIGH MONTHLY COSTS for such a high end circuit. Now while some people have Fiber that supposedly supports "1GB/1GB Download-Upload", in truth that is just to the 'first connection point' like your local service providers neighborhood or city-wide equipment center office building. Once it goes from there, again, it depends on the rest of the connection 'between' that point and your destination.
Think of the connections like the Highway/Freeway systems you drive on, and no not everyone can drive 100MPH right from the Garage door to ANY store's parking lot they want (Upload), and back (Download). Besides all the 'traffic stops / lights' and slower zones 'things happen' that cause slow downs and then there is construction and so on that redirects traffic at that specific time your trying to get to the store. Same with the Internet, there is many things that can 'slow' a part down.