To answer your actual question...yes, if you can, stay in the Park Hyatt Shinjuku. I've stayed there four times a year since 2000.
Things to look out for:
1. Two Maybachs normally parked out the front....
2. All the parts of the hotel featured in Lost in Translation with Bill Murray and Scarlett Johansen. If you have not seen it yet, try to before you go...then watch it again after you come back.
3. The long bar on the top floor. Asian men with Western Blond women and Western men with Asian women...(like my wife)
4. The long bar singer. Don't know who it is at the moment but whoever it is, she will be vaguely familiar and will sing forgettable songs. Don't smile at her too much because in the break, after you've had a few, she'll come over and try to get you to buy her latest (non-label) CD...which of course, you will.
5. Stunning service
6. No tips
7. More restaurants than you can eat in.
8. A bill for about £3000 for a week...if you're careful.
Of course, you could always stay at Keo Plaza or Hilton for less cost and just visit.
Unless you know a local, it is difficult to go sightseeing on your own because you just tend to wander about not quite understanding what's going on. Best to take one of the many tours the hotel will offer which are very well run and not expensive. For instance, coach to Tokyo Station, Shinkansen to Fuji including lunch etc around £175 for a day.
I notice you are going in December (planning well ahead!) if it's early December, pm me.