To my understanding, Jonathan Swift's "A Description of a Morning" simply points out the everyday occurrences of the serving and working classes as each day begins.
Swift describes the maids, Moll and Betty. Betty is found leaving her master's bed, going into her own room to mess up the covers of her bed, as if she had slept there. Moll prepares to wash the floor with a mop, showing a fair amount of experience with her "dex'trous airs." Later she sells brick dust, used to sharpen knives. A "youth" (we assume a young man) is using a blunted broom to sweep off the curb; the coal man is selling his wares. The chimney sweeps are working and calling, as the "duns" or debt collectors gather at "his lordship's gate" wanting to be paid. Swift even gives the reader a glimpse into the common practice of bailiffs (sheriffs/jailors) who released their prisoners for a short time to (ironically) steal money (breaking the law, while in jail—we assume—for already doing so) in order to pay their jailers for special privileges.
The line in question begins this slice-of-life insight into a normal London day.
Now hardly here and there a hackney-coach
The first people on the job, it would seem, are the carriage drivers, though there are only a few at this early hour. A "hackney-coach" is a means of transportation for hire. Whereas those of wealth would own at least one carriage with driver and livery, many other people used this form of public transportation. This is where the practice of hiring a "hack" would have come from—a term we hear today in a more modern setting. While it meant to hire a carriage then, today it refers to a taxi—a car for hire. Today's drivers need a hack license in order to drive a cab.
There is the remainder of the quote provided:
...showed the ruddy morn's approach...
This seems to mean that with the appearance of the hackney coaches (or carriages) morning has also arrived.