This week's book giveaway is in the OCAJP 8 forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide and have Edward Finegan & Robert Liguori on-line! See this thread for details.
You'll need a custom log appender. Tomcat normally uses the java.lang.utilities logger (juli), but I'm not sure what support there may be for syslog using that logger. The Apache log4j logger has a syslog appender, however you have to build a custom version of Tomcat to use log4j. Instructions are included in the basic Tomcat documentation.
The syslog appender literally does what it says, which is to log to the OS syslog facility. I'm not actually sure what it does when the OS is Windows, however, since Windows doesn't have syslog, it has an event recorder. On a Linux machine, logging syslog to a remote logger would be done by configuring the OS system logger itself to record to a remote system.
That's the basics. A little googling might direct you to something more suitable, such as a log appender that formats syslog network packets and sends them directly. I don't know if one exists, but there's no reason why it shouldn't.
An IDE is no substitute for an Intelligent Developer.