Taxi is best way to get around Dubai. Cheap and plentiful. Ask the hotel to book one for you in the morning if it makes you feel better.
Hotels. Plenty to choose from within 10 minutes of the airport. Back in the day when flydubai paid for the hotel we stayed at the Samaya Hotel Deira. Breakfast was excellent and on a quiet street. Another option would be to stay at the Pullman Hotel at City centre mall. A few people I've met stayed at the Premier Inn Dubai Airport. Only thing about the PI is that it is on a busy road, don't know how quiet it would be during the night. Doesn't matter where you stay as you can always take a taxi to any place in the city.