Eilat Airport website, with all relevant information: http://www.iaa.gov.il/Rashat/en-US/Airports/Eilat/
Eilat Flight Booking can be done at the following websites:
Eilat is a small city, and the Eilat Airport is within a 10 minute walking distance from the Eilat hotel strip. However, if you wish to make your ride comfortable, a taxi would be the best option for you. The taxi ride should cost approx. 25 NIS from anywhere in Eilat.