The better solution is to have your cron job write to a log file.
Then, you can run
But, what you propose is likely possible. You'd need to "wrap" your existing script with code like this, below. This gets a handle to the GUI session, and then launches gnome-terminal, running your command.
# This is your new cron script.
dbus_address=$(cat /home/$user/.dbus/session-bus/*-0 | \
grep -v '\#' | \
grep BUS_ADDRESS | \
cut -d '=' -f 2-)
# Here, replace /path/to/script.sh with whatever the old cron script was.
su $user -c gnome-terminal -e "sudo /path/to/script.sh"
Anyway, there are still a lot of issues with this. Please don't try to do it; you'd be better off looking at Nagios or some other monitoring software.