The Croce del Sud was built in 1933/34 designed by the famous Milanese architect Carlo Enrico Rava.There is more about Rava in this article:
http://nicholaskilner.com/designer/carlo-enrico-rava/
I'm not so sure about the dating of the Casa del Fascio. We may need to do more research on this. What I have so far is that it was inaugurated as the headquarters of the Fascist party in 1938 as the picture below shows.It could have been built earlier than that . At that time it was the biggest building in Mogadishu. It later became the first parliament of Somalia.
A note on Carlo Enrico Rava, he was the son of Maurizio Rava who was the governor of Italian Somaliland from 1931 to 1935. Maurizio Rava from Milan, was a prominent member of the fascist party, a decorated general, a painter and writer. The surprising thing about him is that he was, alongside Guido Jung and Aldo Finzi, one of the famous Italian Jews who gained prominence in fascist Italy. Fascist Italy paid lip service to anti Semitic and many Jews thrived during this period, although many also suffered due to German pressure on Mussolini.
The literary and artistic side of Maurizio is evident in Mogadishu where in 1993,he reconstructed the old Sultan of Zanzibar ' s residence and converted it into the famous Museo della Garesa, the Somali national museum.