It is unlikely that the entire Klink-2 Computer Science Ontology is openly available online.
The authors decide to monetize their research creating Rexplore.
It seems that the generated CSO ontology is a very substantil part of the project.
The demo of Rexplore is available here. Following instructions, one can generate topic trees for different series of Springer proceedings. However, the "Download tree" button is not working.
However, a sample of the ontology is available.
The paper you have linked to references to another paper, which contains this link.
The content of the ZIP archive shows how OWL ontologies looks like.
However, those example ontologies (or rather taxonomies) covers Semantic Web topics only.
As for API for ontologies.
If an ontology is serialized into RDF, then you can make queries with SPARQL.
Languages for querying on more abstract level are DL Query and SQWRL.
There even exists the so-called OWL API.
But first, just open an ontology in Protégé.
Update 1
The 2012 ACM Computing Classification System has been developed as a
poly-hierarchical ontology that can be utilized in semantic web
applications. It replaces the traditional 1998 version of the ACM
Computing Classification System (CCS), which has served as the de
facto standard classification system for the computing field.
The full CCS classification tree is freely available for educational
and research purposes in these downloadable formats: SKOS (xml), Word,
and HTML.
Source
Update 2
The Knowledge Media Institute (KMi) of The Open University and Springer Nature are happy to announce the release of The Computer Science Ontology (CSO).
CSO is a large-scale, automatically generated ontology of research areas in the field of Computer Science. The current version of CSO incorporates 14K topics and over 143K relationships extracted by applying the Klink-2 algorithm on a dataset of about 16M scientific articles.
Source
