A day pass for public transport which is good for unlimited travel between ZRH and the city center costs like CHF 14 (that's around USD 15).
You can choose to take the tram or the train into town. The tram station is like a one minute walk from the HR. You can't miss it--you'll pass it as you walk from arrivals to the circle shopping center where the HR is located. The tram takes like 50 min and runs at least 4 or 5 times per hour.
The train station is on the lower level of the airport center, around a 5 min walk from the HR. It runs even more frequently and takes like 12 min or so. Thus, I'd recommend taking the train even though you might see slightly less of Zurich than when riding the tram.
Excellent info. So happy to see this property open up and its location is great. I would have no problem staying here for leisure purposes as the transport links are easy and frequent. I have stayed at the Radisson Blu but found is quite underwhelming. I am hopeful for better from Hyatt.