Compare "da geht die Post ab" (there's a lot going on, usually said of events, not implying chaos). The denotional sense is the place where the post is sent off.
If you've worked in a dispatch center (note dis- ~ off) you'll note that it is quite hectic and initially chaotic. Terminsgeschäft is a matter of timing, therefore everyone is in a hurry. (it is also quite often the middle of the night, but I'm not sure whether that had anything to do with it). The lot is initially unsorted and needs to be sorted out. Hence a connotation of chaos can be gleamed, though a postal master will naturally master it. However, it's no secret that messages have gone lost occasionally.
Similar expressions are "ab geht die wilde Lutzie", "ab geht die wilde Fahrt" which leave no doubt that haste and turbulence is implied. The semantic field includes "da stept der Bär", and the like, which implies an association to carnival (and carroussels, as far as "Fahrt" is concerned). By the way, in the industrial age, post is sorted on a "carrousel", a more or less complicated network of conveyer belts, shoots and ladders, and other machinery like the luggage disposal at an airport.
So the semantic associations of the post should be clear. To explain "Ab die Post", compare
a sole "Ab!", which is also heard quite frequently
Ab! -- Away!, Go!
Incidentally geh, or to go are akin to a preposition, that is Gr ago "towards".
die Post is an appelative. Today it softens the expression, that would otherwise appear rather brisk. Historically, the appelative was perhaps simply denotional, but I have no quotes to assertain that believe. Such appelative is archaic, but still understood: Einen schönen guten Abend, die Damen!.
I'm not sure whether that should be deemed a case of accusative (die Damen, den Herren), which would make some limited sense, or rather nominative (Hallo, der feine Herr). The syntax can be seen as well in "[schneid'] ab den Mist", "Weg damit", etc. "Post" means equally the mail, the personal and the office, so the phrase is underspecified.
We can also compare Hepp! Hop hop, ab in's Körbchen! Wohl an! Auf auf, raus aus den Federn, Ihr Schlafmützen! Raus! Raus mit dir [du] Halunke! Stop, Hammertime! Stehengeblieben! Kompanie, halt! An die Waffen! Alarm, alarm, die Russen kommen! Vorsicht, Taschendiebe! Alle mal aufgepasst! Immer rein in die gute Stube! Geh mit Gott, aber geh, du Nervensäge! Aus die Maus!
Perhaps compare "to go postal" (to go past reasonable, beyond repair).