Rain forecast after weeks of dust — what should I actually check around the villa before the first rain hits?

You've spotted the actual risk: in Dubai it's almost never the rain, it's months of dust turning the first rain into mud and blocking the drains that carry it away. Quick pre-rain round:

• Clear every drain the water relies on. Roof outlets, balcony and terrace gullies, and parapet scuppers collect a surprising amount of sand over a dry summer. A handful of sand plus one blown-in plastic bag is all it takes to back a balcony up until it overtops the threshold and comes inside. Lift the grates and scoop them out.
• Then test them. Pour a bucket of water down each one — if it doesn't drain briskly, it's partially blocked. Far better to find that now than at 2am in a downpour.
• Check thresholds and door sweeps at balcony doors and the main entrance — that lip and seal is what keeps a shallow pool of standing water from wicking indoors.
• Secure loose outdoor items. With gusts forecast around 45km/h, anything light on a roof or balcony (planters, a satellite dish bracket, garden furniture, shade sails) is worth checking.
• The outdoor AC unit is fine getting rained on — it lives outside — but clear leaves and sand from around its base so its own condensate and the rain can drain away.
• The glass gets a muddy film. Dust-laden rain dries into hard mineral spotting on windows and screens; wipe it off before it bakes on in the sun rather than letting it sit.

Where water actually pools against a wall or sits on a flat roof, that's a waterproofing question for another day — note it and get it looked at after. But the pre-rain drain clear-out is the bit that prevents most first-rain flooding, and if you'd rather have someone run the whole roof-and-balcony round properly, it's a straightforward job for a handyman in Dubai.