I'm re-posting some useful location and surrounding facility information that I already mentioned last year when this hotel was still a Novotel. I doubt very much of any this information has changed even now that it's a Hyatt.
Location and surrounding area facilities and transport:
-Right next to the newly built "Airport Hotel" MRT (metro) station. The station number is "A14a" It's one stop away from Terminal 2, and two stops away from Terminal 1. Tickets to either terminal cost around 20 TWD each way...which is about 0.60 euro cents, or 0.60 USD cents. However be aware that the express train that runs between the airport and central Taipei does NOT stop here. You will have to change at either of the 2 airport stations to get the express train. This is very important when ARRIVING at this hotel from the city centre, as if you stay on board the express train after airport terminal 2 it will go right past the airport hotel station.
-There is a FREE shuttle bus to/from the hotel to both terminals. Transfer time is about 10 minutes to terminal 2 and a bit longer for terminal 1. To be honest, it's now probably more comfortable to use use the underground train than the shuttle. More space and better air conditioning. Shuttle is a bit cramped in comparison and I think walking times between the shuttle drop off points and check in desks on the top floor of the airport are longer.
-There is not a single store, bar, or restaurant outside the hotel and there is absolutely nowhere to walk to. If you want anything at all such as a convenience store or a proper restaurant meal outside the hotel, you will have to take the MRT (metro) or shuttle to the airport and back. It's not possible to walk there.
-Windows at this hotel do NOT open for fresh air.
-When this property was a Novotel hotel there was a coin operated laundry where you can wash all your clothes for a reasonable price. It's quite well hidden though and not advertised much. It's located in the basement of next door building which is also part of the hotel. To get there go outside of the hotel by the main entrance and turn right. Walk straight along the outside windows of the hotel, and walk right past the entrance (elevator and escalators) of the MRT station. Do not enter the station at all, but keep going straight on street level. You will eventually arrive at a building entrance with some sliding glass doors, and inside is an unmanned reception desk. Go in and take the elevator on the far right 1 floor down to level B1. You have arrived.
There are 2 washing machines and 2 dryers. It costs 60 TWD for washing 1 load and soap/detergent is automatically added. Cycle lasts 30 minutes exactly. Drying costs 10 TWD for every 6 minute increment. Calculate at least half an hour for proper drying which means a minimum of 50 TWD for drying. Both the washing and drying machines only accept 10 TWD coins so make sure you have enough of these, or get proper change at the main reception before heading over. Make sure you pick your stuff up at soon as it finished, as otherwise there's a very good chance your clothing will be unceremoniously thrown in one of the available plastic laundry baskets (even if you are even a few minutes late) by another guest who needs one of the machines as well. Chinese/Taiwanese people don't like wasting any time.
I hope Hyatt keeps this coin operated laundry.