When we lived on the west coast, the only options
DCL gave us were to arrive the day before, or take the red-eye. The price was the same, either way, so we always opted for the day before. On the page in the brouchure which describes the air package, there's a list of cities and symbols which will designate whether or not they'll require you to travel the day before or overnight. There's no charge if you're flying from one of these cities.
We found the Hyatt so convenient that we've continued to book it, even when we purchase our own air, which for the past couple of years has been much less expensive. On our last two cruises, we've flown across the Atlantic, arriving Friday afternoon. Again, we booked the Hyatt on our own.
Check-in for the busses is usually at around 9:00-9:30am. The first bus leaves around 10:00. If you're staying at the Hyatt, let them know you're a DCL passenger when you check in, and they'll give you some information about the transfers, explaining when and where to check in and meet the bus.