Where is the best country in the Middle East to start as a fresh grad in the computer industry?

Jordan.
Jordan is known as the region's top tech hub; it would definitely make a great starting point for any fresh grad in the computer industry.

Struggle will make you create a room in any area

Jordan

Living and working abroad is not for everyone, but if you have a touch of the pioneering spirit and a skill to offer, you could do worse than spend a few months or years in the Middle East, a part of the World that is riding out the Global recession better than most.
As for a Fresher in the computer indusrty UAE is a better option for a starter.

This question is hard to answer, but easy to find by your self "use bayt.com!" Simply, setup a job alert for the position you like to fill and start analyze your daily emails.