Looking into the water problem has led me basically to a combination of the Nabatean Cisterns (which wouldn't work well on their own for obvious reasons), and a primitive form of desalination - probably more useful as salt extraction.
Essentially you have two wood-faced (or stone-faced) channels, over which hangs a tilted roof, with a wall on the lower side of roof - this would force the roof to be either north or south facing so that sun can hit the water all day. The Salt Channel is on the outside, just under the roof, but with the roof angled in such a way that the sun can still hit the salt water. The water evaporates upwards to the roof (probably wood as well - depends on what there is to hand), which drains towards the wall, that drains into the second channel. The saltwater channel connects to the coast, or more likely the Creek (or some larger canal in the future), whereas the freshwater channel drains into an underground cistern that is used as a water source.
Might be best to have it doubled back to back to exploit more sunlight with two salt channels around a middle channel - but this would probably provide some water, and allow the nascent settlement to export sea-salt. This could be the work of our
Arabian Archimedes. In fact, I have the idea that this would evolve to be a raised water system on top of houses, but that would require a water wheel and an Archimedes screw at the very least.
So yeah, to make this work looks like we'd need an early development in hydrodynamics - which if it was a specialty of the town, would probably have a larger impact than just making Dubai livable at this point.
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So using that for salt and water - and fish and any imports (and maybe dates from some sort of functioning agriculture?) Dubai would have access to pearls for trading, and perhaps import/export work. It isn't the most resource rich location. Perhaps goods processing (Import Gems, Export Jewelry?). It could work as a ship construction yard, but all the wood would have to be imported for obvious reasons.
All of this requires stubbornness and dedication (sort of like when Dubai exploded in recent history).
Further afield, in Ras Al Khaimah, there are decent clay and limestone deposits - which could be shipped by sea, the clay more than the limestone, which opens a window for pottery.
For any level of survival, let alone building power, I'm guessing it would have to work to get on good terms with the various Bedouin tribes in the interior, it may be vital as if they can form a partnership with them then Dubai can benefit from the raiding loot and caravan taxation, then again, that could be said of most settlements at this time. Perhaps the early modesty of Dubai might spare them the raids that Yemen or Oman (who would be more likely to provoke the Bedouin), would suffer - perhaps the full agreement that caravans from Dubai will pay their 'tariffs' into accounts in Dubai for the various tribes could give them a level of safety that other caravans may lack.
As a coastal settlement (and after a huge amount of digging/flooding) - it could take advantage of salt-tolerant crops - of which there are a surprising number that seem to be fine with salt-water, or high-salinity

Apparently there is a variety of spinach that is salt-water tolerant, and a large number of grasses. The diet of the city would be rather unusual compared to its neighbours - perhaps
@John7755 يوحنا is right, a cult could be useful, especially if it has a thing for eating salt-water crops, would certainly stimulate propagation.
The next big option would have to be trying to get a hand in on trade - this isn't that easy as Persia could do much of their trading themselves - so Dubai would need to fiercely compete to get a toe in the door - perhaps shipping incense in from Yemen, and Silks from the far east over sea.
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Prospective brief TL
-> POD (1) Arab trader from a tribe near Dubai goes to India, where some of the saltwater crops grown there fascinate him, and he decides to bring some home with him, growing them on Dubai Creek. There isn't anyone or anything of value there, so by and large they are left alone.
2) The crops grown there attract more people to copy him, leading to a small agricultural community - and perhaps a cult around the 'miracle' of saltwater crops. Large village, perhaps traders come to trade goods and pearls.
3) A member of the community travels with Bedouin caravans to the Med, learning from Greeks, and others (think a small-scale Ibn Battuta), seeing salt production in other areas - basically boiling water.
4) The traveler comes back, and basically figures out a primitive way to desalinate water using wood/canvas and trenches - and uses it both to expand the arable area of their crops, and for fresh water.
5) The now small farming community has access to a larger water supply, and is now relevant enough for people to notice - using the Creek and the coast as a mooring and trade stop, the fresh water and curious food makes it notable.
6) People in the town start buying and selling goods, to shorten trade journeys to and from Persia, making a small bit of money - and some goods start being processed.
at this point we have a moderate town, which likely has nearby coastal villages mimicking their agriculture - this forms the beating heart of a small state which mixes things up in the region. Now it just comes down to whether it can compete with the other Arab communities to build up a force to start waging war - perhaps allying with or paying the Bedouin. Combining the tricks of hydrology, Arabian wind towers, and the salt-water crops, it would certainly change coastal Arabia significantly.
Sea, Salt, and Sun - An early of history of the Malharab State