Nonny's Post office is the Shop segemnt from the episode: Happy Clam Day!
Nonny opens a postal service. His first customer, Molly, has come to mail a package when Nonny informs her she'll need stamps first, the more it ways the more stamps it'll need. She reveals that the box is filled with pink feathers, so it will be light. Nonny puts it into a machine and it reveals it will need just 2 stamps. And so, Nonny puts on the 2 stamps before Oona brings him her package. She reveals she's sending a watermelon, which he think it will be heavy and put it onto the stamp machine. It turns out to need 9 stamps! And so upon stamping it, Oona leaves.