Take the tram nr. 28 and forget all those overpriced rides with the "sightseeing tram". The daily metro ticket also works by the way and it costs only 6€ for all the busses and subways in the city.
When you are up there take your time to enjoy the view. It is full of history. The guide (a paper) that you can get in the front desk will give you a first insight of the rich history of the castle. Don't forget to go to the little museum in the castle. It is easily overseen but provides great information about the place itself.
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