Does anyone read the part 2 of Nagarjuna's karika as an argument about infinite divisibility? Is Nagarjuna is arguing that time cannot really be both infinitely divisible and extended, so is a conceptual construction?

I'm asking because verse 2.23 suggested that to me, taken completely out of context, anyway.


  
  [A going] which is other than the going by which a goer is made evident does not [enable a goer to] go.  Because it is impossible for
  going to be twofold within a single goer.
  


Emphasis, obviously, mine. I really like the idea that Nagarjuna was saying that time must be infinitely divisible, but time as we experience it (evidently) is not infinitely divisible, because if it was it would take forever to compose it.



As I think it was pointed out in an answer, it is often I think said that Nagarjuna means time cannot inherently exist because it is infinitely divisible. I'm just adding that it can neither be infinitely divided nor not.  

I would not conclude that time does not flow, but that anything that arises has already passed, radical impermanence, suggesting that extinction and so on is already present.
Here is the translation that by Jay L. Garfield of that same verse:


  2.25 One that both is and is not a goer does not go in any of the three ways. Therefore, going, goer and that which is to be gone over are non-existent.  


This is from the second chapter of Nagarjuna's Fundamental Treatise where Nagarjuna refutes the inherent existence of motion: going and that which goes. This verse is the last of the chapter and Je Tsongkhapa in his Ocean of Reasoning explains (by adopting Buddhapalita's earlier explanation) the three ways refer to:


That which is gone 
That over which one is going
That which over which one has yet to go


ie., the three times. Je Tsongkhapa says there is another way to understand the three aspects here (by adopting Chandrakirti's explanation) as referring to:


That which is a goer
That which is not a goer
That which is and is not a goer


In sum, the verse can be read as refuting the inherent existence of the goer, the place one is to go, or the going. All of these exist as mere conventions. There is no essence of motion.

Anyway, maybe you can update your question to indicate how you arrive at the conclusion that time cannot be both infinitely divisible and extended? That's basically saying that time cannot be regarded as modeled by the real numbers... but rather must be the infinite integers?? I don't think this is what Nagarjuna is saying because even if time is infinitely divisible and extended it would still be proper to model time by the past, present and future. 

Let's say that the past, present, and future are modeled by real numbers. All we have to do is identify the present with a specific real number and say all those numbers less than, but not equal to the present model the past and all those numbers greater than, but not equal to the present model the future. Even with this, Nagarjuna's argument in the chapter would still accomplish the goal: refuting the inherent existence of the goer, the act of going, and the place where one goes...

You also refer to verse 23 in your update, but this verse according to Je Tsongkhapa is just a refutation to the idea that there is more than one action or "going" that makes a goer go. Nagarjuna's position is that only one action is tenable because a two-fold action makes no sense. Although it is obvious that it takes time to perform an action, these verses and this chapter do not explicitly deal with time. The chapter is about motion and how motion does not inherently exist.

However, there is a chapter in Nagarjuna's Treatise that explicitly deals with time. That is chapter 19 and there is one verse in particular that deals with whether or not time "endures":


  19.5 A nonenduring time is not grasped.
  Nothing one could grasp as time
  Could exist as enduring.
  If time is not grasped, how it is known?


You can read more about this verse and Je Tsongkhapa's explanation in Ocean of Reasoning, but I would just add that it is clear from the first phrase that we experience moments of time as enduring. That is directly contradictory to the way I used the real numbers above to model time. We perceive the present moment as enduring for some length of time. If we look at the present moment and analyze it we'll find it is made of parts. For anything that has an extent must be made of at least three parts: the beginning, the middle, and the end. Thus, if the present moment is enduring then it must be made of parts. If it is made of parts, we can analyze further and ask if an inherent entity that is this present moment can be found in the beginning of this present moment? In the middle ? In the end? By looking and analyzing like that it can be known for sure that the present moment cannot be made of any inherent entity. The present moment does not exist inherently.

So, I'd conclude and say that it is precisely through analyzing an enduring moment of time and breaking it up and dividing it infinitely we can know that no enduring moment whatsoever truly exists. Time does not truly exist. It only exists as mere label. It only exists conventionally.

That is emphatically not to say that time does not exist at all! That way lies nihilism and we can be sure that if we look at the above explanation as somehow indicating nihilism, then we have grasped the snake of emptiness incorrectly and ought to put it down right away!