1) No you do not necessarily need an internet connection to work with Neo4j (Although you will need it to at least download it).
Neo4j can run in two modes, embedded or server. When running in embedded mode, you just need to instruct/configure your code to point at the location of the directory which houses the database, whilst in server mode you will need to be able to access and communicate with it via its HTTP REST API. Depending on where you have installed the Neo4j server, you may or may not need an internet connection. By default, the Neo4j Server is bundled with a Web server that binds to host localhost on port 7474, answering only requests from the local machine, so for local development this is easily accessible, just point a browser at http://localhost:7474/browser and you are away. Otherwise it will more than likely be within some kind of private networking setup. If you don't need an internet connection for your network then this should be fine too. The book has a chapter dedicated to running Neo4j in these two different modes which provides more info on this.
2) Cypher is not a standalone installable tool, Cypher is the graph query language used to interact with Neo4j (a little bit like SQL for the RDBMS except this is for graphs You can use Cypher from within a number of tools including via the Neo4j console or shell which comes for free when you download and install Neo4j server.
If just want to get a feel for Cypher without installing Neo4j, you can visit the online Neo4j console which provides a way to get some insight into Cypher and the Neo4j world here: http://console.neo4j.org
Thanks a lot Nicki. one thing that i noticed is when i open neo4j,it does not ask for username and password like relational databases do.Is there any way to make neo4j prompt the user to enter username and password.It will be more secure that way,right?