At first, Tsukasa's initial shyness around Nasa, despite being married to him (as well as her reserved nature), seemed to just be due to her personality. However, it makes sense after the discovery of her being immortal. She is over 1400 years old and grew up in a period where relationships and social cues were much, MUCH stricter than those of today and, from her perspective, society has only recently eased up on them.
Tsukasa's fondness of Nasa's name becomes this after her origin is revealed. She has spent centuries looking for a way to contact Kaguya, and had been hopeful when NASA managed to send men to the moon. While she was sad that she still could not get in contact with Kaguya, she was still amazed by one of humanity's greatest accomplishments and still is to this day.
Tsukasa being a Supreme Chef also makes sense after the reveal. She claims that she got better by practicing... and she has had about 1400 years' worth of practice! No wonder she is practically a pro, since she must have learned those skills since she had to do SOMETHING during all that time, along with her many other skills that she must have sharpened over the centuries.
Tsukasa's adoration of modern forms of entertainment(anime/manga/movies) and her indifference towards historical places also make sense in hindsight. She has been around for over 1400's years and she LIVED through those events, so historical things aren't interesting to her but she is amazed at the novelty of modern things like anime and manga, which is new and exciting for her.
For most of the story, at a first glance her behavior is quite mysterious and bizarre, considering how she just unconditionally trusts Nasa for little to no reason (handwaving it as "I'm a good judge of character"). She also seems to be casually dropping hints about herself having lived for several centuries with a total disregard of whether Nasa figures it out or not, however she never states it outright. This is because she is unaware that Nasa - due to his injury-induced amnesia - has forgotten her already revealing this fact during their first meeting. She thinks he already knows that she's immortal and remembers how he won her heart with his promise. Her behavior suddenly makes far more sense and is less bizarre in hindsight when all of this is revealed.
When you start to grasp exactly what Tsukasa's infinite regeneration entails. She is shown to slice off her own arm in order to escape pursuit with it growing back seconds later. She then admits to going to great lengths to try and die and testing these abilities. While not explicitly mentioned imagine the horror of a body regrowing a decapitated head (or worse, the reverse), crushed by giant boulders, drowning, being burnt alive, etc.
