I would do in this way.

./server > /tmp/server-log.txt &
sleep 1
while ! grep -m1 'Server is active' < /tmp/server-log.txt; do
    sleep 1
done

echo Continue

Here -m1 tells grep(1) to quit at the first match.

I veryfied my answer with my toy "service" below:

#! /bin/bash

trap "echo 'YOU killed me with SIGPIPE!' 1>&2 " SIGPIPE

rm -f /tmp/server-output.txt
for (( i=0; i<5; ++i )); do
    echo "i==$i"
    sleep 1;
done
echo "Server is active"
for (( ; i<10; ++i )); do
    echo "i==$i"
    sleep 1;
done
echo "Server is shutting down..." > /tmp/server-output.txt

If you replace echo Continue with echo Continue; sleep 1; ls /tmp/server-msg.txt, you will see ls: cannot access /tmp/server-output.txt: No such file or directory which proves the "Continue" action was triggered right after the output of Server is active.

For me to read service's status as to service app:

$ /sbin/service network status
network.service - Network Connectivity
   Loaded: loaded (/lib/systemd/system/network.service; enabled)
   Active: active (exited) since Ср 2014-01-29 22:00:06 MSK; 1 day 15h ago
  Process: 15491 ExecStart=/etc/rc.d/init.d/network start (code=exited, status=0/SUCCESS)

$ /sbin/service httpd status 
httpd.service - SYSV: Apache is a World Wide Web server.  It is used to serve HTML files and CGI.
   Loaded: loaded (/etc/rc.d/init.d/httpd)
   Active: activating (start) since Пт 2014-01-31 13:59:06 MSK; 930ms ago

and it can be done with the code:

function is_in_activation {
   activation=$(/sbin/service "$1" status | grep "Active: activation" )
   if [ -z "$activation" ]; then
      true;
   else
      false;
   fi

   return $?;
}

while is_in_activation network ; do true; done

Use grep -q. The -q option makes grep quiet, and it will exit immediately when the text appears.

The command below starts ./some-service in the background, and blocks until "Server is active" appears on stdout.

(./some-service &) | grep -q "Server is active"