I've been using RDBMSes since college and am really struggling with the underlying concepts of NoSQL databases...but I think their concept is really cool.
I believe I understand the following (please correct me if I'm wrong, because these play into my question here!):
AFAIK, Hibernate is an object/relational mapping framework that only supports SQL-like databases. So you won't need / be able to use it if you use Neo4j, you would use Neo4j's API instead.
But nothing prevents you from using both Neo4j and an SQL database therefore mixing Hibernate and the neo4j API (most likely to store/query different objects within your project).
Have you checked the basic examples given on Neo4j website, such as http://docs.neo4j.org/chunked/snapshot/tutorials-java-embedded-hello-world.html ?
You are right, NoSql does not define a specific standard. You might want to have a look at this (short) introduction: http://martinfowler.com/articles/nosql-intro.pdf