What does ‘Lower oneself for farewell swim’ mean? Is this a sport related jargon?

I like MT_Head's answer, but there were two things I wanted to suggest about the interpretation that wouldn't easily fit into a comment. I absolutely agree that this piece is heavily metaphor-mixed and confusing.

First

After avoiding the muddy pool of his boss, James L. Dolan, for the duration of his time as the Knicks’ president

could mean that Walsh did not get "caught in the quagmire" represented by his boss. A quagmire is boggy ground (e.g., a muddy pool) that will trap your feet when walked on. This expression refers to getting into a situation that is difficult to get out of. Given Dolan's reputation, this could mean that Walsh handled Dolan's ego and management style successfully, without any major conflicts during his tenure.

Also, it is possible (although a stretch) that this could refer to "muddied waters" between Walsh and Dolan. When we "muddy the waters," we introduce elements that make a choosing an appropriate action unclear. For instance, "Walsh managed his players well until Dolan muddied the waters by hiring a team manager."

So the phrase

Walsh refrained from lowering himself for a farewell swim

then means essentially that Walsh didn't ruin his reputation before he left by getting into a nasty, name-calling, finger-pointing fight (as MT_Head said) or by confusing things for the team.

Second

He took the same high road out of Madison Square Garden that he rode in on.

I'm particularly fond of MT_Head's colorful "horse you rode in on" expression, but I think this confused metaphor actually refers to a different horse. When someone is morally upstanding, especially to the point of looking down on others, we say they are "on their high horse." You might hear the expression "get off your high horse," which means "stop being so judgmental and hypocritical!"

This makes more sense in the context that MT_Head gave: He left with the same appearance of high moral standards that he started with.

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?)