Vienna was the capital of an empire of 50 million people until 1918, and then, almost overnight, the oversized capital of a small country. It never quite shrank to fit. The Habsburgs ruled from here for more than 600 years, and they left the palaces, the art collections, the opera house and the habit of doing everything with a bit of ceremony. Then came Vienna 1900, when Freud, Klimt, Mahler and Otto Wagner were all working within a few streets of each other, and the darker 20th century: Hitler's years here as a failed art student, the Anschluss of 1938 and ten years of four-power occupation after the war.
None of that makes it a museum city, though it has enough museums to fill a month. Vienna regularly comes top of the world's most livable cities, mostly thanks to things visitors barely notice: cheap public transport, a quarter of the population living in good city-owned flats, and vineyards inside the city limits. Five days lets you do the imperial sights properly and still have time for the other Vienna, the one that sits for two hours over a single coffee and ends the day with young wine at a Heuriger.
Day one is the old center and the Habsburgs, day two the great museums and the opera and concert quarter, day three the Belvedere, Hundertwasser and the Prater, day four a trip up the Danube to the Wachau, and day five Schönbrunn, Beethoven's Heiligenstadt and a farewell evening in the vineyards. It's also the best introduction to Austria as a whole, since half the country's history was decided in these streets.