The public transport in Eilat is very well developed. Practically every area in and around Eilat is accessible by bus. The bus lines are operated by Egged Transportation Ltd, a company that offers various tickets such as one-way, day and week tickets. Please note that the buses do not run during Shabbat! At the central bus station (12 Sderot Ha-Tmarim, Eilat) you can get additional information and also tickets.
The website of Egged is: www.egged.co.il
If you want to save yourself the trouble of always having to buy single tickets, you should take a look at the smartcard “Rav-Kav”. This is a card with which you can load a certain credit or a certain type of ticket (e.g. weekly ticket). Further information is available at: www.ravkavonline.co.il