The Hyatt nr Taksim Sq is the Grand Hyatt.

The easy way to do it is to take the Havas (pronounced like "havash") from the airport to Taksim Sq. The Havas is a very efficient and cost-effective airport shuttle bus. It has a few routes, the one you want goes, of course, to Taksim Sq. When the Havas drops you off, the driver will point the way to the hotel. It's literally around the corner and when you walk a few meters ahead you will see the hotel roof on the right. If you have more than carry-on, you probably want to take a taxi, but I took the Havas and stayed at the same hotel and it was, clean, efficient, perfect and very inexpensive, and you can see lots of interesting things on the ride.

http://www.havas.net/en/

After you make it through arrivals, just ask for the Havas. It was on an island on the arrivals level and the busses are clearly marked. Ask for the one going to Taksim Sq.

You'll need some Euros or NTLs. I think it's 10TL. ATM inside the airport after exiting arrivals/customs.

If you're concerned about the short walk from Taksim to the hotel late at night, then I guess you will need to either take a taxi or the hotel car from the airport, but 250 Euros for the hotel car is too expensive.

-David