You can easily achieve this with an initscript (script file under /etc/init.d/), using LSB tag "Required-Start" with "$network" parameter:
### BEGIN INIT INFO
# Provides: After network script launching
# Required-Start: $network
# Required-Stop: $network
# Default-Start: 2 3 4 5
# Default-Stop: 0 1 6
# Short-Description: start a network dependent script
# Description: Network dependent script launcher
### END INIT INFO
Use the skeleton here is you haven't your.
Then modify the "do_start()" function to launch your script, for example:
do_start()
{
# Return
# 0 if daemon has been started
# 1 if daemon was already running
# 2 if daemon could not be started
cd /my/script/dir && ./myScript.sh
# Add code here, if necessary, that waits for the process to be ready
# to handle requests from services started subsequently which depend
# on this one. As a last resort, sleep for some time.
}
You should modify the "do_stop()" function too to avoid possible shutdown errors.