What is happening to Nasa here when Tsukasa does something adorable?
As Aki Tanaka has said in the comments, this is most likely the Spit Take trope.

A Reaction Shot to an unexpected event that involves spraying food or, more commonly, a drink from a character's mouth. ... A common Lampshade Hanging for this trope is when a character pauses to take a drink just so he has something to spit out in reaction.
(emphasis mine)


This trope is also used for comedic purposes, as seen in the image of Ryuuji Takasu above. This trope is also used when someone is surprised or excited, similar to the nose bleed trope that is commonly found within anime and manga. You've said that Nasa Yuzaki reacts that way whenever Tsukasa does something he sees as cute. It is possible that this spit take trope is being applied the same way a nose bleed is.
Source