Debian - how to run a script on startup as soon as there is an internet connection ready for use?

Put your script in /etc/network/if-up.d and make it executable. It will be automatically run each time a network interface comes up.

To make it do work only the first time it is run on every boot, have it check for existence of a flag file which you create after the first time. Example:

#!/bin/sh

FLAGFILE=/var/run/work-was-already-done

case "$IFACE" in
    lo)
        # The loopback interface does not count.
        # only run when some other interface comes up
        exit 0
        ;;
    *)
        ;;
esac

if [ -e $FLAGFILE ]; then
    exit 0
else
    touch $FLAGFILE
fi

: here, do the real work.

This is a job very well suited for systemd.

Running a script as a systemd service

If your system is running systemd, then you can configure your script to run as a systemd service which provides control over the lifecycle and execution environment, as well as preconditions for starting the script, such as the network being up and running.

The recommended folder for your own services is /etc/systemd/system/ (another option is /lib/systemd/system but that should normally be used for OOTB services only).

Create the file e.g. with sudo vim /etc/systemd/system/autossh.service :

[Unit]
# By default 'simple' is used, see also https://www.freedesktop.org/software/systemd/man/systemd.service.html#Type=
# Type=simple|forking|oneshot|dbus|notify|idle
Description=Autossh keepalive daemon
## make sure we only start the service after network is up
Wants=network-online.target
After=network.target

[Service]
## here we can set custom environment variables
Environment=AUTOSSH_GATETIME=0
Environment=AUTOSSH_PORT=0
ExecStart=/usr/local/bin/ssh-keep-alive.sh
ExecStop=pkill -9 autossh
# don't use 'nobody' if your script needs to access user files
# (if User is not set the service will run as root)
#User=nobody

# Useful during debugging; remove it once the service is working
StandardOutput=console

[Install]
WantedBy=multi-user.target

Now you can test the service:

sudo systemctl start autossh

Checking the status of the service:

systemctl status autossh

Stopping the service:

sudo systemctl stop autossh

Once you verified that the service works as expected enable it with:

sudo systemctl enable autossh

^{NOTE: For security purposes systemd will run the script in a restricted environment, similar to how crontab scripts are run, therefore don't make any assumptions about pre-existing system variables such as $PATH or whatever you have in your /.bashrc, ~/.zshrc, etc. Use the Environment keys if your script needs specific variables to be defined. Adding set -x at the top of your bash script and then running systemctl status my_service might help identify why your script is failing. As a rule of tumb, always use absolute paths for everything including echo, or explicitly define your $PATH by adding Environment=MYVAR=abc.}

The internet connection is brought up by an entry in /etc/rc6.d/ probably S35networking. If you change that file and insert your commands at the end, or better add a /etc/init.d/mystuff and link /etc/rc0.d/S36mystuff to it and insert your commands there, then that will start as soon as the network is up.