Can I use the desktop version on my localhost? What settings need to be made for this? I am using Node js as a server
You can certainly download and install Neo4j desktop. It runs on Windows, OS X, or Linux.
Node js is a separate VM from Neo4j Desktop and the Neo4j instances you start from Desktop.
I mean, can I connect to Neo4j Desktop from an express js application and control it as if I did it with a server database?
Yes if the the express js application is running on the same system as the DBMS started in Neo4j Desktop.
You would use localhost for the URL and the user name would be neo4j, the password would be whatever you set up when you created the DBMS in your Neo4j Desktop project.