As a Swedish citizen, you may enter Japan without a visa for up to 90 days; proof of an onward or return ticket may be required.
You could leave the airport during your layover. After passing through immigration, you can proceed to the train, which will be the fastest route to central Tokyo (there is also bus service, taxis, etc...). Both the Narita Express and the Skyliner serve the airport. You'd likely want the Narita Express, but your exact route would depend on where you're going. A round-trip ticket is available for 4,000 yen, and the trip from Narita to Tokyo Station takes roughly 53 minutes, running every half hour. Consult the timetables for exact times. There are also packages including public transit tickets available.