There is the central library, it is free and open to the public but not all of it is open shelf though.
It's website states:
Literature research - how to get my book?
Our current stocks hold about 6 million items most of which are in closed stacks and have to be ordered in advance.
General and subject specific reference works as well as ...
The Berlin City library (ZLB) is different than the State Library (Staatsbibliothek zu Berlin) that you mentioned. The English pages of both should give a general idea of the conditions of most public libraries. For the Staatsbibliothek, fees have been abolished for the next 2 years, but registration is still needed.
As a general rule, if a book does not ...
In this page you have the airport information. It seems that you need a receipt but maybe you can call to one of these numbers and they can find a solution of you:
Terminal 1, between Concourses B and C, Level 2 (Departures), open daily from 6 a.m. to 10 p.m., phone +49 69 690 73277
Terminal 1, Concourse B, Level 1 (Arrivals), open daily 24/7 hours, ...