sheikh zayed road is not a bad place to work - you might be able to find accommodation there if you're willing to share.
is the aed 36000 your housing allowance or everything included (p/m or p/y)? if it's your housing allowance, you stand a pretty good chance of getting something decent (once again, if you share, unless you are willing to put money into your rent from your own salary).
i've been told a good area to look at are bur dubai, mankhool, golden sands, sheik zayed road, satwa (certain spots are safe, others not). there are more, but i'm not an expert
hopefully someone with more info. can help you out. just don't take anything that is cheap - it's bound to be in an unsafe area. also check out gulfnews.com - classifieds. that'll give you a good idea of what is what.
rent is VERY VERY VERY high here at the moment. i was blessed enough to find a place in springs (to share and closer to jebel ali than dubai) - even with this, i'll still have to put in some money of my own toward expenses etc.
p.s. congrats on getting the post