For anyone curious about the rakugo in the animated shorts, they're modern retellings of classic rakugo stories.
https://www.youtube.com/watch?v=_iAS5LDik2s&feature=youtu.be
In the Toki Udon/Toki Soba one, which is the most popular due to the animation, Bunko acts as a person who heard of a Rakugo story called Toki Udon, where the character was able to pay less for his meal by counting halfway then asking about the time, then continuing his counting according to the time rather than how much he had initially counted, thus paying less coins than required. The person then tries to use the same trick at another store, but the store clerk replies "Sorry, but we only accept e-payment."
https://www.youtube.com/watch?v=o0cCwiASlh0
In this, it's a retelling of Manjuu Kowai. In the original, Character A is asked what he's scared of by Character B. A replies that he's extremely scared of manjuu but they are actually his favourite treat. B then tries to scare A using manjuu, but A instead gorges himself happily. In this one, Bunko's first character says that they bought tapioca (bubble tea/boba) to scare the other person. The other person then says "Ah, I'm so so very scared. So scared that I can't upload it to social media." The first character then realises they got tricked, and asks what the other person is truly afraid of. After a short silence, the latter says they're afraid of Likes (as in likes on social media.)
There's another modern day retelling on the author's twitter, but it's buried pretty far down and I don't remember what it was other than it was based on Okiku's Plates.
Unfortunately though, it seems like the rakugo in the actual manga are the regular tellings rather than these modern versions from the special animations.