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This type of question can generally be answered by translating the term into the local language, then searching for it.
Launderette is Vaskeri in Norwegian according to Google translate. Searching for Vaskeri Arendal has various results including a business directory: http://www.nettkatalogen.no/bransjelist/vaskeri%20arendal/1.htm
And just to confirm there's not a translation error, searching for the address of one of those vaskeris gives the following image in Google street view http://goo.gl/maps/VeYWK (zoom in to the store behind the parking sign). Looks like a place to wash stuff (rather than a store selling washing machines).
Of course, there is no guarantee it is still there.