An easy way to redirect all stdout/stderr of a given process to syslog is to do the following in a bash script:

exec 1> >(logger -t MYTAG) 2>&1

Please note this only works with bash as this is a bash extension to the POSIX standard.

If you want only stderr:

exec 2> >(logger -t MYTAG)

So if you have an application that runs in the foreground (and a properly written service should not daemonize in my opinion), you can have the following shell script for a quick solution:

exec 1> >(logger -t MYTAG) 2>&1