No cheap on an absolute basis, but cheap for Dubai
There are hundreds of one-star hotels, mostly in Deira. Figure on about 180 dh (US $50) single/ 250 double. For that price you will get a private room w/ bath, AC, TV, refrig, etc. (Avoid the guesthouses, which are basically workers' dorms.)
Forget about being about to book them over the internet, and plan on paying cash (no cards). There are accomodations desks at arrivals, but they have little or no interest in helping you find a budget room.
The airport bus (401) leaves every 30 minutes around the clock. (Out arrivals, turn left, and walk 100 yds) Take the bus (3 dh, try to have coins or small bills) to Al Sabka station (where everyone gets off). You will be smack in the middle of things, a very short walk from the Gold Souk. You will see hotels on every street in every direction. A good first choice is the Sima directly behind the station. If they are full, keep trying. You will find something, though the cheaper/better value places go first.
The hostel charges 80 dh for a bed, 170 dh for a single, and is located a long way from anything.