After searching some 7000+ pages of the Posthorn, only a few articles on Arendal & Christiania appear. It is noted that only a few specialized catalogs list these locals.
Norges. Byposter og Annen Norsk Lokalpost. By Carl A. Pihl. The Stamps of the Private Byposts of Norway. S. Sanness and H. R. Holmes. Katalogisierung der Norwegischen Lokalposten (Bypost). Jurgen Tiemer, 1984,
In particular, the top one by Pihl appears to be the most authoritative. A copy recently sold for under $5
I came across this interesting notation (translated)
"Many of the last city posts (and smallest) sold the bulk as collectors' stamps. The sale of stamps to philatelists was also a prerequisite for financing city mail in small places with a limited customer base, and that activity was similar to today's CTO? services. "