Post A Letter? Jokin' Aren't Ya!
Let me tell you about Post Offices in Dubai. There's one in Jumeirah. There may or may not be one in Satwa - last time I passed it, it looked like they were demolishing it. There's the main one in Karama. No doubt there's one or two in Deira. All of these sites have actual post office boxes, where you could collect your mail if you were fortunate enough to be able to get a key to one of them. I've also seen clumps of P.O. Boxes at some gas stations. There's a mini-clump at Media City. Anyhoo, don't want one, never going to get one. My mail goes to a shared Media City box, and if you are dumb enough to snailmail me stuff, I'll pick it up once a fortnight and then throw it away.
As for sending mail, I usually do it via Media City also. There is another alternative - our local Choithram's sell stamps and they have a Post Box just outside. If your package seems a little bit heavy they weigh it on the fruit and veg scales and it comes back with a sticker that says 'Brown onions, 53 grams, 3 dirhams'.
If you are unfortunate enough to receive something 'suspect', then you will get a card ordering you to go to a real Post Office to retrieve it (Samurai Sam has a hilarious account of such a visit - but I can't find the link...). This is Old Dubai at its worst, and if snail mail wasn't such a total irrelevance I would be jumping up and down about it.