Less than two hours away from Copenhagen center
http://www.visitdenmark.com/funen/attractions/attractions-within-two-hours-eurovision-island
Less than one hour away from Copenhagen center
http://www.visitdenmark.com/copenhagen/attractions/denmark-attractions-less-one-hour-copenhagen
Some suggestions where to have a byte or a drink, and relax
https://maps.google.com/maps/ms?ie=UTF&msa=0&msid=216177005688550166485.0004fdaef88eb9fe6922c
(legend : Søernes ølbar - nice bar with a large beer choice
Nørrebrogade - lots of good kebab places
Skt Hans Torv (+ nearby Elmegade) - nice restaurants
Nyhavn - quite famous area, nice bars
Grøbrødretorv - centre of town, nice restaurants
Ved kanalen (Christianshavn) - some good restaurants
Christiania - very interesting area, google it )
World cup open air screens:
http://www.trippy.com/place/53b26c24e4b00f1957322caf