The prince and princess arrived at a performance AT the Royal Theater in Copenhagen.
"At" is used to indicate the place where something happens, or is situated. It does not take into consideration whether the place is indoors or outdoors. Here is the first entry for "at" from the American Heritage Dictionary:
"In or near the area occupied by; in or near the location of: at the market; at our destination."
Here is one reference to a performance (of something) AT a theater, from Google:
... A group of talented actors performed the musical theatrical performance of "Oliver" at the Mabel Tainter Theater this past weekend. ...http://www.mabeltainter.com/ _______
You might say IN the theater if you are referring only to the space inside the actual theater, as in this ad to rent the same theater, from Google:
"¢ The Mabel Tainter Theater has a variety of opportunities for rentals. We have the place for you - whether you would like to get married in a Victorian setting, have a reception in the Bertha Tainter Room, hold a business meeting or company retreat, or present a band concert or other performance in the theater. To check availability or for more information, please call 715... http://www.mabeltainter.com/rent.htm
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