Take a look at the Holiday Inn, Port of Miami. You can get a room that overlooks the port so you are able to see your ship in the morning.
Taxi ride from the airport is set rate of $24, for up to 4 persons- then you can taxi to the pier in the am for around $8. It is opposite the Bayside Marketplace, so lots of shops, bars, restaurants all set around a Marina.
If you take a shuttle, they charge around $15 each to downtown hotels, but you have to wait and make rounds of other hotels before you get to yours.