There is not much I can say about this that would help to make it any better, or any worse for that matter.  This, by far, is one of the most professional pieces I have seen on this site thus far, so congratulations!  I really did enjoy the descriptive narrative and the lack of dialogue was a bonus as well.  A few things...

It fell in heavy drops against the office window of the 12-story building and trickled down the glass in chaotic currents before passing from view to the streets below.

As you can see, I have underlined the parts of this sentence that bother me, and really, it is very minor.  The use of so many 'the' words, yes, only four, makes it almost on the edge of a run-on.  It's not really, but for some reason I have to empasize that particulat word every time it crosses the path of my eye and it seems to make the whole sentence drone on.  Maybe you could split this sentence into two?  Just a suggestion...

It was nothing like Arkansas. No trees. No slow moving rivers. No shady afternoons with friends on porches sipping pleasure from the rounded lip of a shared mason jar.

It was New York.

I know this passage is meant to reiterate just how not like Arkansas New York really is, but you need not state that it was New York again.  Perhaps you should consider deleting it from the first paragraph and leaving this realization to this point in the story.  The reader already knows it is New York and does not need to be told twice.

A half eaten pastrami sandwich sat abandoned on his desk top next to a box of cold McDonalds French Fries.

He went two differrent places to get lunch?  Unless McDonald's has pastrami sandwiches now, I don't know.  Just seems a bit odd.

A gold letter opener and a neatly sliced envelope topped the organized mayhem that accompanied the lunch.

 

I'm sure this is intentional, but don't the two kind of cancel each other out?  I understand the picture, and it does work.  So I guess leaving it be wouldn't hurt anyone.

At the front of the desk in a brown mahogany humidor, the sweet smell of Havana’s finest circulated with memories, secrets, and promises made in passion. A giggling intern; her panties at her knees. A rare almost solitary moment. An impulse. The deep moan of a young girl surrendering to boyhood charm. Within the box a single cigar, dark and pungent hid among the other less important hand-rolled decoys. The top of the box was adorned with a presidential seal and a small gold plaque that read, “A good cigar is like a memory; shared intimately, sparingly and anticipated slowly like a guilty pleasure.”

 

I love the descriptions here.  Perfect ending.  But is it meant to be as dirty as it sounds to me?  What exactly does the cigar have to do with anything?

Well, those are my thoughts.  Keep it up, great writing!

 Excellent.  

I disagree with the critique that said "It was New York" was unnecessary.  I think that sentence, with its own paragraph, contributes to the mood of the piece with a kind of repetitive, almost poetic, rhythm.  However, I would like to see that first paragraph reworked to play up and point to that terminative statement.   

A half eaten pastrami sandwich sat abandoned on his desk top next to a box of cold McDonalds French Fries. Under the sandwich and covering the desk were the scripts and speeches for a half-dozen scheduled speaking events. A gold letter opener and a neatly sliced envelope topped the organized mayhem that accompanied the lunch.

Since I am reading this for rhythm, I will suggest some changes to this paragraph.  Remove the crossed out words, and change "the" to "his".  Also hyphenate "half-eaten".  I like "organized mayhem".  

The envelope was addressed simply to “Kitten” The address reflected a trendy Greenwich Village fashion boutique. It was postmarked in New Jersey with an Upstate New York P.O. box for a return address. The letter was stamped impersonally with the red-faced declarative “return to sender.” 

This is a bit confusing, about whose address is where.  I use a mail forwarder myself, but don't understand how the letter was returned to Harlem.  Maybe you could make this a bit more clear.

The now indistinguishable message contained on the letter wasn't threatening, nor sordid. It was simple: Time heals all wounds. It was a pass phrase, like “the coast is clear” or “ali-ali in come free.” It was an invitation.

How about -- "was neither threatening nor sordid"?  I prefer "time wounds all heels," but that's neither here nor there.  Your phrase is appropriate.  In the children's game, I have always heard the phrase as "Allie, allie out's in free."

Altogether, a good read.  I enjoyed it very much, and would like to see more of your writing.  Maybe a bit longer next time?

Very witty--it didn't hit me until the description of the single cigar.  Then all the other clues started to fall into place--Arkansas, the McDonald's fries, the Harlem office.  I think it might not have occurred to some of the other readers who your protagonist actually was, and the effect was really nice, almost as if he's masked in some way.

Loved it--subtle, and quite humorous, a very sophisticated piece!

 I usually say that I liked or loved a work at the beginning of a critique but in this case I'm more apathetic to it.  The writing style on the other hand was brillant and hard to ignore.  And again with the New York sentence/paragraph, it was a bit much.  The single sentence paragraph was a good premise, but the line was a little redundant.  Onto my other little notes though.

It fell in heavy drops against the office window of the 12-story building and trickled down the glass in chaotic currents before passing from view to the streets below.

I felt it necessary to take note of one part of this sentence that really bugged me.  It wasn't the four the's.  Acutally it's the phrase trickled down the glass in chaotic currents that caught my eye.  The sentence was great until that phrase.  The word trickled seems to drag down the sentence when you use chaotic currents only a few words later.  If you're using something strong, like chaotic currents, you have to use a strong verb as well.  Trickled makes the whole thing seem as if the rain is weak and light while chaotic currents gives the impression of strength or maybe power rather.

My only other thing I want to comment on is actually the second sentence.

The hot and notoriously oppressive New York humidity bowed momentarily to the baptism of cool healing rain.

This line was glorious!  I aboslutely fell in love with it.  The word choice glowed as weird as that sounds.  That is the kind of line that makes people want to read a story.  There are little snippets that show that kind of word choice throughout the rest of the story, but here is where is shines the best.

Overall, this is great -- obviously your writing is solid and professional, as everyone here has been saying.  That said...this felt like the beginning of something to me.  The plot, in and of itself, didn't even happen until the last few sentences.  I would love to see the character and his inner conflict introduced a bit earlier in the piece.  For the first half or so, New York itself seems to be a character -- and while it would be fascinating to see a piece on Bill's relationship with the state where his wife is a senator, that doesn't seem to be what you're going for in this story.

Also, it seems like you're working off some really great metaphors which never quite get taken advantage of -- the description of the weather, the "organized mayhem" of his desk, the fact that he is surrounded by so much excess of food, none of which he is really interested in.  All of these are amazing, and you start to get to them but never quite exploit them to their fullest degree.  You could definitely make those more explicit, and really milk them for all they're worth.