So, according to documentation all logging functionality can be manipulated in logging.properties file which is usually located under apache tomcat root installation folder, which in ubuntu case is under /etc/tomcat6/…
Opening a file you’d probably spot logging level settings (set to FINE as default), well We want to get rid of logging at all, so, according to this helping post I needed to change ConsoleHandler level to OFF.
java.util.logging.ConsoleHandler.level = FINE
java.util.logging.ConsoleHandler.level = OFF
And happily I got the result Apache Tomcat6 solr server stopped logging to catalina.out