Peter Levi's translation of The Guide to Greece by Pausanias, from the second century. This is a detailed travel guide that describes all the art in the temples, the agoras and acropolises, and gives a good idea of what you will see at public events. He tells ancient legends, folktales that are really pretty modern, and lacks only maps. His descriptions are so good that you can often draw up the city center from him.

The best source I can think of is Juvenal's Satire 3. It basically gives a bunch of reasons why no one should live in Rome. Most of his satires talk about life in the city but this one is about the city in general. Other poets like Catullus, Ovid (Amores) and Martial also give a good indication of "regular life".

Yes, there are many.

Starting on page 251 of this book you'll find Herodotus's description of Babylon, written in the 4th Century BC.

Alexandria was founded in the 3rd Century BC, and both Arrian of Nicomedia and Diodorus Siculus both wrote about the founding of the city. Though their writings were centuries after the fact, Arrian did have access to texts that were later lost, including Ptolemy's biography of Alexander. (Ptolemy was Alexander the Great's most trusted advisor.) Maybe not the "vivid" description you're looking for, though.

Probably more up your alley is the description of Alexandria in "De Bello Alexandrino" written circa 40 BC and attributed to Julius Caesar or someone close to him. Here is an English translation of the original Latin text.

I'll leave it up to others who may know more about Rome.

The problem with many of the texts of ancient historians is that their writings aren't always matter of fact. Depending on the author, they frequently stray into "tall tale" territory. For instance, in Herodotus's description of Babylon, he describes the city's walls as being ~50 meters thick, with a hundred doors. Later research reveals the city had more like eight doors in the city walls. So take these kinds of accounts for what they're worth.