Is it possible? Of course it is; almost everyone does it, in a form or another.
How to do it well, now, is a whole other question.
Edit: Just to be sure, here I'm talking about interleaving whole paragraphs, pages or even chapters text.
I'd personally avoid interleaving unless it is strictly necessary. It may be just my opinion, but if I'm reading a story, and the author stops every five paragraphs to illustrate me some aspect of the world, there will be two possible outcomes:
- Either the writer is so good that is descriptions are actually interesting to read, or
- I will be bored the hell and back.
The main con of interleaving is that you stop your plot. If, by chance, you had the reader's attention, you risk losing it with something completely different and seemingly out of place. As you said, it's the exact definition of clunkiness. If that's not bad enough, you risk giving the impression of "feeding" the information to your audience - with a process sometime referred as "infodumping" - and that's usually fastidious.
So, mix it up
The best option here (again, imho) seems to try to embed whatever you need to explain about the context in the pace of the story. Of course, the pace will go down, but that's not necessarily a bad thing.
Of course it isn't so easy as it seems. You still have to become good at it to avoid breaking the action too much or infodumping. As all things in writing, it needs practice and perfection may as well be unattainable, but whatever.
The tricky part, indeed, is mixing contextual explanations in the story flow.
I'm not mentioning "introspective" parts, because those are almost natural in comparison. It's pretty normal to pair up descriptions of what happens with description of the feeling of a character, especially - as in your example - with a first person narrator. Good practice too, since it is a good way of making the reader acquainted with your character.
So, are cycles ok?
Well, probably. I'm a bit unsure of what you mean with cycle; I hope you don't literally mean that one "part" as to follow the other in order. My advice is this: write your story trying to give the pace that you prefer, mixing narration with introspection, and when the time feels right embed whatever little context is needed.
There are other strategies around and someone will sure explain them better than me. You could - for example - give a single paragraph of meaningful context at the start of every chapter. Or come up with whatever "structure" you can imagine. It depends on wheter it makes sense with your story.
As for your proxy example, the first paragraph works for me, but the second one is too context-heavy.
Made it! Panting and sweating i found a window seat. I had to make it, really. The service i was on only runs every two hours. Iterations of optimalization to the train schedule meant that now urban areas have ultra frequent connections while the further one travels from the overpopulated cities, the scarcer the transport. A realistic and predictable policy that nevertheless further erodes the already crumbling public and other services in more remote areas.
Do your reader really needs to know that - or rather, to read you explain that - to enjoy your story? I frankly doubt it. You overdid it in the last sentence; a simple comment on the character view about the train sistem policy would have been better. Remember, if you meld the pieces well enough, there will be no visibile separation lines.
Last useful tip: I remember reading somewhere that the audience is not stupid. Yea, that may come as a surprise (joking, ofc). If you have the urge to explain everything in detail, remember that. Most people can grasp the meaning of what your trying to convey without you writing an essay.
On the other hand, you can't expect the readers to be as well informed as you about your story. You will always be more knowledgeable of them on that front, period. Finding a good pace also means accepting that fact and let your audience join the dots to form their own view of the plot.