"Same-day delivery" is one of the least precise phrases in online selling. It can mean anything from two hours to eleven at night, and the gap between what your customer assumed and what you meant is where most complaints start.
Agree the cut-off with your courier
Every delivery promise depends on a cut-off: the latest an order can come in and still go out that day. Agree it with your courier first, then publish it - not the other way round.
Distance matters more than you think
A delivery across town and a delivery to another emirate are different jobs. If you sell across the UAE, consider different promises for different areas rather than one promise you cannot always keep.
Promise a time, not a phrase
"Delivered by 9pm if you order before 2pm" is a promise your customer can plan around. "Fast delivery" is not.