There are multiple hints that the unnamed city the plot takes place in is a reminiscence of Genzano di Roma. Ende has lived there during the writing of Momo, and the Italian background (the names, but also the ancient amphitheatre) hint at that. Apparently, Ende himself has acknowledged the influence in an interview, but I could not find the original interview.
„Das ist für uns ein Geschenk, eine große Ehre“, betont
Gabbarini. Die Geschichte über das Mädchen, das es mit den Zeitdieben
aufnimmt, verfasste Ende in seiner Wahlheimat. Die Berge um den 25
000-Einwohner-Ort, insbesondere der Monte Giove, seien seine
Inspiration für die Schildkröte Kassiopeia gewesen, sagt der
"It is a gift, and a great honour", stresses Gabbarini [major of Genzano di
Roma]. The story about the girl who combats the Men in Grey
has been written by Ende in this residency of his. The mountains
surrounding the small village - most of all Monte Giove - were
inspirations for the tortoise Kassiopeia, said the major.
News Coverage of a festival dedicated to Michael Ende in Genzano di Roma
Regarding when it takes place, there are no hints at that and I haven't found anything Ende has stated about that matter.
To me, it feels like the 1950s. I haven't lived during that time, but the rise of capitalism and the modern monetary system was most present in Germany during the Wirtschaftswunder, and Ende acknowledges Momo being a criticism of modern economics.
Furthermore, the open smoking and the prominence of cigars is something attributed to the 1950s in German literature - as far as I know. An entire subplot of Homo faber by Max Frisch, set in the 1950s is concerned with a tobacco plantation.