There is a large and very efficiently running community on facebook - called "Cats in qatar". They do a lot of good stuff and help a lot of people. it is also kind of a discussing place for pet problems and suggestions.
Post your offer there - there are many people who require pet sitting and post about it on CIQ.
For dogs, another community called dogs in qatar , also exists - though i do not know much about it.
you can check that out too.
Good luck. :)