Haneda to PH Tokyo?
 

Is there a limo bus from Haneda to the Park Hyatt Tokyo?

How about a train?

Which is best?

Next day leaving hotel to Narita?

Narita Express from Shinjuku or bus directly from hotel?

I don't know what time you arrive and what time you leave, and the following may sound like heresy to many (including myself), but I'd stay somewhere else, close to Tokyo station (e.g. the FS Marunouchi at Tokyo Station). Then you could take a train or bus to Tokyo station (or even a taxi, if you can stomach the 5-6,000 yen it will cost you from Haneda), and then the NEX next day from Tokyo Station. The PHT is all the way across town, and I'm not sure that the bus from Haneda to the PHT (if there's such a thing, actually) runs with any decent frequency.

The Friendly Airport Limousine bus travels from both NRT and HND to the Park Hyatt.

http://www.limousinebus.co.jp/e/

The NEX train will get you to Shinjuku Station from NRT and you can catch the hotel shuttle (I believe there is one) or a cab from there. I imagine there is train service from HND to Shinjuku Station, as well.

My advice is to avoid the train service from HND to/from Shinjuku station, unless you have light luggage and you know how to navigate the train system. I tried it last year and needed to transfer once while navigating the train system. It was a lot of hassle.
Take the limo bus for direct drop off.
There is no NEX from HND to/from Shinjuku to my knowledge.

There is no JR service at Haneda. If you want to do rail: Take either the Keikyu train to Shinagawa or the monorail to Hamamatsucho and in either case transfer to the Yamanote line to Shinjuku. Get a taxi to the Park Hyatt.

I was surprised, given that Haneda is much closer to Tokyo than Narita, how long it took, practically as long as the Narita express, although the trains are more frequent.

I don't think that you would save more than about 15 minutes by switching to Tokyo station, as Pickles suggests. But it is true that staying near Tokyo station would help you get to Narita faster.

If you do take the train, buy your Narita Express ticket for the next day when you arrive at Shinjuku or Tokyo, wherever you are staying.

For the Park Hyatt and a first time visitor under these conditions, I'd recommend the limousine bus in both directions if the schedule from Haneda meets your schedule. Note that you can take the bus to the Shinjuku Washington hotel and the Park Hyatt is only about 200 meters walk, however, the bus is still not that frequent. There is extremely frequent service to Shinjuku station though, and the taxi to the Park Hyatt is only Y660.

Note that if you want to take the Narita Express and want the Park Hyatt to get your ticket, you should write them in advance or they will charge you about Y1200 for taxi fare to/from Shinjuku station to get the ticket for you if it is very short notice (like next day).