The cost of shipping your belongings will vary depending on the amount of containers and from where the cargo will ship. For example. one full 20ft container shipped from New York to Oslo will set you back about 800USD port-to-port. You can have a read through our page on shipping and removals for some more information.
Are you quite attached to your current belongings? Depending on your lifestyle, you could save a little by buying inexpensive furniture from places like IKEA, considering you will be in Norway for five years, which might seem long, but goes by fast.