Hauptbahnhof hotels near this train station that offer free airport transfers

I would personally stay a mile or less from the train station if you would prefer to walk and ride. That being said if you have an average budget I would choose a place that offers complimentary breakfast. You will thank me later. I have travelled all over the world and Zurich is absurdly expensive. I am talking $16 for a value meal at Mcdonald's expensive. $5 for a cup of coffee expensive. For a point of reference, I did not find Paris expensive at all. Also, 3 days would be the absolute most time I would ever spend in Zurich. It is incredibly dry and non eventful. It has some high points...But very few.

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I don't know

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6:00-21:00 - 365 days a year

You may try Arlette Beim HB, Leoneck Swiss Hotel, Royal Hotel Zurich. They all have AC facility in rooms and will fit in $200 budget.

Just walk in!

before considering taking the train on your journey, you have to list first your priority. as for me: 1. i dont prefer train as it takes longer than airplane. 2. the rate of the train is almost same rate of the airplane 3. not much to see while on train. 4. make a comparison between train and airplane if you are travelling on a tight schedule or if you want to maximize your trip, i would suggest go for airplane rather than train.