<sup>**Now solved in full after helpful pointers in comments from @GarethMcCaughan and @Deusovi - my thanks to both!**</sup>

The completed crossword should look something like this:

>! [![Completed crossword, using all English letters][image1]][image1]

Or, alternatively, in the spirit of this month's [Topic Challenge](https://puzzling.meta.stackexchange.com/questions/7406/monthly-topic-challenge-4-cross-non-words):

>! [![Completed crossword, using Japanese characters in place of some two-letter units][image2]][image2]

What's happened here? Well, it turns out that in order to fit all of the words into their spaces, we have to...

>! ...replace some two-letter units with their equivalent **Japanese characters**. Here I have chosen to use [hiragana](https://www.unicode.org/charts/PDF/U3040.pdf) rather than katakana (if this is the wrong thing to do culturally speaking, I plead ignorance - I knew next to nothing about Asian alphabets before ending up deep into solving this puzzle...).  
>!  
>! Note that in some instances (marked in red characters in the first diagram) the same symbol (or a variation on the same symbol) can be used to represent multiple different two-character units, e.g. the TI and DI in the very first box. In these instances I have chosen to portray the simpler of the two variations in the diagram.

In fact, we need to go a little further, and also apply this to the four single letters indicated in grey...

>! ...since these individual vowels also have their own Japanese characters:  
>!  
>! [![Completed crossword, using Japanese characters in place of some two-letter units and vowels in grey][image3]][image3]

We now need to take note of the poem at the top of the puzzle, because this is giving us a very important clue (thanks to @Deusovi in comments for this part)...

>! It turns out that there is a very famous Japanese poem called *[Iroha](https://en.wikipedia.org/wiki/Iroha)*, which famously is a *pangram* that uses **all 47 characters of the Japanese syllabary exactly once** (NB with the exception of ん (-n), which was not distinguished from む (mu) in writing until the early 20th century (Source: Wikipedia)):  
>!  
>! 以呂波耳本部止  
>! 千利奴流乎和加  
>! 餘多連曽津祢那  
>! 良牟有為能於久  
>! 耶万計不己衣天  
>! 阿佐伎喩女美之  
>! 恵比毛勢須  
>!  
>! And this should point us towards the fact (noted by @GarethMcCaughan in comments) that each of the characters used in the crossword grid appears just once. In fact, we have used 40 of the 47 characters and only seven remain unused. If we strike off all those that have been utilised up to this point, we are now left with:  
>!  
>! **KA, KI, I, TSU, SU, YO** and **TO**  
>!  
>! [![Used syllables][image4]][image4]

Finally, for the landmark represented in the crossword, we can now see that this is:

>! **Tōkyō Sukaitsurī** (a.k.a. 'Tokyo Skytree') - a *neo-futurist* building (hence the title) in the Japanese capital:  
>!  
>! [![Tokyo Skytree][image5]][image5]  
>!  
>! This ends in the 'RI' present in the grid in answer 2A, and we are able to arrange the seven unused syllables into these boxes to spell out its name, ending in the 'RI':  
>!  
>! **TO-KI-YO SU-KA-I-TSU-RI**  
>!  
>! The numbers specifically refer to the order in which they appear in the pangram: 2 TO, 6 KI, 4 YO, 7 SU, 3 KA, 1 I, 5 TSU.  
>!  
>! [![Absolutely completed crossword][image6]][image6]

A full list of my clue answers is available here, with available grid space enumerated in square brackets:

>! **Across:**  
>! 2. Dormant Javanese volcano and abandoned nuclear power plant site (5) **MURIA** [3]  
>! 4. Walking with _________, first in a series of documentaries (9) **DINOSAURS** [6]  
>! 7. Mercury crater named after a 19th-century American novelist (9) **HAWTHORNE** [6]  
>! 10. Barry _________, loser of the 1964 US presidential election (9) **GOLDWATER** [7]  
>! 12. Indian alternative medicine system (8) **AYURVEDA** [6]  
>! 14. Soap opera that debuted on the same day as Channel 4 (9) **BROOKSIDE** [6]  
>! 16. Direct current generators using commutators (7) **DYNAMOS** [5]  
>! 18. ______ Hill, female Australian composer (6) **MIRRIE** [5]  
>! 20. White Star Line ship, later Gülcemal (8) **GERMANIC** [5]  
>! 22. "The better wow", per Urban Dictionary (5) **WOWIE** [3]  
>! 23. Small, often domesticated lizard (6) **IGUANA** [5]  
>! 24. Follower of Network or British (4) **RAIL** [3]  
>! 25. Squirrel's nest (4) **DREY** [3]  
>! 26. Nose pimple caused by excessive drinking (3-7) **RUM-BLOSSOM** [9]  
>! 27. Constellation containing the Spare Tyre Nebula (4) **GRUS** [3]  
>!  
>! **Down:**  
>! 2. Pondered deeply (5) **MUSED** [3]  
>! 3. Touch by means of a mutual border (4) **ABUT** [3]  
>! 4. Yes album before The Yes Album (4,3,1,4) **TIME AND A WORD** [9]  
>! 5. Wood-cutter after the tree has been chopped down (5) **SAWER** [3]  
>! 6. Enyaq manufacturer (5) **SKODA** [3]  
>! 7. Pointed end of a hook, or a fish caught by it (4) **BARB** [3]  
>! 8. Anatomical homologue of the human ankle in a horse (4) **HOCK** [3]  
>! 9. Genus of duckbill eels (12) **NETTENCHELYS** [9]  
>! 11. Escher lithograph featuring a compound of three cubes (9) **WATERFALL** [8]  
>! 13. Saint Paul's role as author of Galatians, Romans and others (8) **EPISTLER** [7]  
>! 15. Component of an internal combustion engine pushing the intake/exhaust valve (6,3) **ROCKER ARM** [7]  
>! 17. viz. (6) **NAMELY** [5]  
>! 18. _______ Aouragh, Dutch anthropologist (7) **MIRIYAM** [5]  
>! 19. Genus of the black-winged kite (6) **ELANUS** [5]  
>! 21. South Korean League of Legends player currently playing for Rogue (7) **MALRANG** [5]

  [image1]: https://i.sstatic.net/f7tyH.png
  [image2]: https://i.sstatic.net/XY4Ai.png
  [image3]: https://i.sstatic.net/fMy41.png
  [image4]: https://i.sstatic.net/MNkyF.png
  [image5]: https://i.sstatic.net/GbYVZ.png
  [image6]: https://i.sstatic.net/RcqzR.png