There are some badly mixed metaphors in that passage...
If one has always maintained higher standards than one's peers, one might be tempted to "lower oneself" - to indulge in the same sort of bad behavior as they do.
"Muddy pool" is, itself, a bit of a stretch - to "sling mud" at your opponents means to insult them or defame them; the French have a phrase nostalgie de la boue (a yearning for mud), usually used to refer to the public's appetite for reading gossip. I believe the writer is saying that James L. Dolan is so fond of slinging mud at his enemies that he keeps a pool of it for convenience.
That being said, one wouldn't usually want to swim in a muddy pool! The writer is simply saying that, since he's leaving the job anyway, Welsh could have given the sort of bitter exit speech that so many public figures do - full of slander, accusations, insults... That would have been his "farewell swim" in the muddy pool that his boss keeps.
Answer: no, not a popular idiom at all, and not likely to become one.
The last sentence "He took the same high road out of Madison Square Garden that he rode in on" is also a bit of a jumble. To "take the high road", once again, means to hold yourself to a higher moral/ethical standard - however, it doesn't go with "rode in on." That phrase is most commonly heard in the insult "Fuck you, and the horse you rode in on." The writer may be (indirectly) suggesting that Welsh was strongly tempted to use that phrase in his farewell speech, or that he's had to put up with a lot of abuse (possibly including that phrase.) However, I suspect that the writer wasn't suggesting any such thing: I think he was simply in a hurry and needed to fill his column.
One more note on "take the high road" - the most famous appearance of that phrase, in the song "Bonnie Banks of Loch Lomond", actually has a very different meaning.
Oh! ye'll take the high road and
I'll take the low road,
And I'll be in Scotland afore ye;
But me and my true love
Will never meet again
On the bonnie, bonnie banks of Loch Lomond.
In this song, the person who's going to "take the high road" is about to be executed - a very different meaning indeed! (There are differing theories: is it the "high road" because he'll be in Heaven, or because his head is going to be on a pike above the high road?)