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Om Xubuntu, setting up tomcat7 is a simple command: "sudo apt-get install tomcat7" to download and install the webserver and any prerequisites. On other systems, use the build-in package manager (yum, portage) to install tomcat7 or install it manually using the useful instructions on the pacakages packages website. By the completion of the installation, you should be able to run tomcat and see the "It Worked!" screen when you visit http://localhost:8080
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Using either the pre-packaged binary .WAR file or building it from the source files (using the helpful and informative instructions), the result should be a .WAR file. Copy the .WAR file to the tomcats root directory <CATALINA_HOME>/webapps. Start/restart tomcat and http://localhost:8080/<fcrepo name><fcrepo>/rest should display the fedora repository front page. (I've called it <fcrepo> as its the name of the .WAR file minus the .war extension and it can vary on build number/release.)
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1) Obtain and build the fcrepo-oaiprovider source code
git clone clone https://github.com/fcrepo4-labs/fcrepo4-oaiprovider.git
cd fcrepo4-oaiprovider
mvn package
2) Move the binary to the fcrepo libraries
cp target/fcrepo-oaiprovider-4.0.0-beta-04-SNAPSHOT.jar <CATALINA_HOME>/webapps/<fcrepo>/WEB-INF/lib
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