By Taxi: You can take a Taxi directly from the airport. The taxi fare is around 250-300 AED and it takes 1 hour 30 minutes.
By Public Transport: From the Dubai International Airport (DXB), you can take the Dubai Metro to Al Ghubaiba Metro Station (the metro ticket is around 5-6 AED). There is a big central bus station there. You need to take the RTA bus E201. It will take you from Dubai to Al Ain Central Bus Station and it takes about One hour 45 minutes (105 minutes). The bus fare is 25 AED. The bus departs every 30 minutes from Al Ghubaiba bus station. Once you arrive at the Al Ain central bus station, you can take a taxi to your hotel.
For more details about timing and schedule of buses from Dubai to Al Ain, check the RTA website (https://www.rta.ae/wps/portal/rta/ae/home?lang=en).
Option 1: If you take a flight from Etihad Airway, it offers a complimentary coach service to Al Ain directly from Abu Dhabi Airport ( https://www.etihad.com/en-ae/experience-etihad/etihad-express/ ).
Option 2: You can take a taxi directly from the airport. The taxi fare is around 350-400 AED and it takes 1 hour 30 minutes.
Option 3: You can first take a taxi to Abu Dhabi central bus station. Then you need to take bus X90. It will take you from Abu Dhabi to Al Ain bus station in 2 hours. The bus fare is 25 AED. The bus departs every 30 minutes. Once you arrive at the Al Ain central bus station, you can take a taxi to reach your hotel.