The Park Timna is a famous natural experince about 25km in the north of Eilat. We were there and are sure that this place is worth a cache.
Please find here more informations about the park.
http://igoogledisrael.com/2012/11/timna-park-the-ultimate-desert-destination-in-israel/
The park is always open for hikers and cyclists.
If you arrive with by car or need further information, the information desk is open in the following times:
Sunday - Thursday ans Saturday 8.00 - 16.00
Friday - Holidays evenings 8.00 - 15.00
in July ans August Sunday - Saturday 8.00 - 13.00
Good luck!