Sam Leiter gave me a copy of a summary of the plot in Japanese (below) which I had summarised by u/Extension_Pipe4293 on Reddit:
A playboy called Mamushino Jirokichi disguises himself as a toolmaker’s assistant and breaks a famous Totoya teacup. To apologize, he pretends to commit suicide by jumping off the Ryogoku Bridge but is saved by Hanagaki Shichisaburō. Despite being saved by Shichisaburō, Jirokichi then deceives him and his sister, Otsuyu, and steals 50 ryō from them.
When Uwabami Kyūta learns of Jirokichi’s true identity, he follows him and tries to extort money from him, so Jirokichi stabs Kyūta. Surprisingly, Otsuyu falls in love with Jirokichi and runs away from home to be with him. Jirokichi asks Shichisaburō for Otsuyu hand in marriage but as Otsuyu has already been arranged to marry another man, through Kishida Yoshikazu’s, mediation, the arrangement is about to be broken off. Jirokichi soon learns that Otsuyu’s intended is Jirokichi’s benefactor and that Yoshikazu is his real father. Jirokichi reforms himself and tries to commit a joint suicide with Otsuyu but is stopped again by Shichisaburō.
In the end, Jirokichi decides to turn himself in to the authorities to pay for his crimes and vows to marry Otsuyu upon his release from prison.
u/Extension_Pipe4293 finishes by saying “It was dramatized from 三題噺 the impromptu rakugo story based on three subjects, Totoya tea cup, drowning oneself, little cuckoo.”
One thing to take from this is that the “fish shop tea cup” in the title isn’t just a tea cup from a fish shop. Actually the “Totoya Chawan” is a type of tea cup that is highly valued in Japan, modelled on a cup presumed to have been invented by the owner of a tea shop in Korea in the 1580s. This type of cup is made to this day and artisans study for a long time to be able to craft the intricacies of this type of cup.