This chapter bothered me a lot, so I went and read the raw. It makes things a bit clearer I think. For those interested...
This is how I'd translate some of the lines from shortly after he woke up on p17, which mimic the initial style on p6: "Burned by a dream, the flower of life's petals fell while filled with regret." "I thought the end would never come." "Surely this journey would just be one of wandering while piling up nothing but regrets." After that, the lines have a slightly different speaking style (-no as a suffix for the "I had a dream" line, for example) which make it even more clear that it's Tsukasa reflecting on how her "long journey" ended.
So the boxed lines on p6, along with the boxed lines on p17 onward after Nasa wakes up, are actually thought by Tsukasa. Probably, since she saw Kaguya last chapter, it triggered some old memory for her, so she also had a dream that morning. Only the boxed lines on p7-8 (when he starts wandering around the dream) are Nasa's thoughts. Hata deliberately hid the transition from Tsukasa to Nasa back to Tsukasa.
Not sure who "burned by a dream" refers to, but since it implied they died full of regrets, it's not Tsukasa herself. Perhaps it's Prince Shoutoku, whose role Nasa played in the dream according to other commenters who know history better than me. In any case, I view this as a happy chapter, since Tsukasa's smile at the end is her way to say "I'm happy now that my journey is over". So this chapter has nothing "new" thematically, it just carries on the message of chapter 102 where Tsukasa talked about how she got married because people don't live forever... she's suffered a lot of tragedy because she's had to see people die around her over the years. People are speculating that Nasa is the reincarnation of Shoutoku, but I don't believe any clear evidence for that... in fact, I think such a development would cheapen Tsukasa's tragic past. It's very possible that Nasa was somehow seeing Tsukasa's memories and experiencing her feelings, her sadness, near the end, or maybe he was just overcome with sadness at seeing her look so forlorn in the dream.