Locating a specific German Ph.D. Dissertation from 1959
In general,
I'd always expect the university library where the dissertation was done to have at least one exemplar in their archives.
The national library is also supposed to have at least one exemplar of every book published in Germany.
For searching books within Germany, the university libraries of several German Länder have a common online catalgoue (Gemeinsamer Verbundkatalog, GVK), see also https://www.bszgbv.de/ which also include Baden-Württemberg and Saarland.
GVK lists 3 exemplars of the dissertation at the Göttingen University library: https://kxp.k10plus.de/DB=2.1/PPNSET?PPN=146113365
https://www.worldcat.org/ is an online catalogue covering even more libraries from around the globe.
Any online resources may be unlikely for a dissertation that old, but there are some possibilities. Try these.
The Classics department at Göttingen.
The university library at Göttingen.
Proquest, which was founded as "University Microfilms", published a lot of dissertations in that era. This may be a longshot for a German dissertation, but it was used by many universities to "publish" dissertations.
The author is unlikely to still be alive, but it is just possible that one of his students is and can provide a copy, or at least an abstract. If you have a citation of the work, the author who cites it may have a copy.
A good research librarian at an academic library can probably help a lot in such a search.
First, search the library information system of the university to get some basic information about its availability at the university library.
Second, you can access it via the German national library, you can access the book in Leipzig and Frankfurt/Main.