Well, for your particular case, you’d make a script, and a service to run that script on boot. Once the service starts, it will keep itself alive.

Here’s the script:

bluetooth-reconnect.sh

#!/bin/bash

dbus-monitor --session "type='signal',interface='org.gnome.ScreenSaver'" |
  while read x; do
    if echo $x | grep -q "boolean false"
      bluetoothctl connect A1:11:22:3A:CD:F1
    fi
  done

You’d place this script somewhere that has system execution privilege (if your distro uses SELinux). I’d recommend using mv for this instead of a file explorer. I will use the directory /usr/scripts/ for example purposes (note that you will have to create this folder, perhaps with mkdir /usr/scripts). Make sure to mark it executable with chmod +x /usr/scripts/bluetooth-reconnect.sh

You’d then write a service to start at boot, just really barebones and simple:

bluetooth-reconnect.service

[Unit]
Description=Reconnect Bluetooth after waking from sleep
After=default.target

[Service]
Type=simple
ExecStart=/usr/scripts/bluetooth-reconnect.sh

[Install]
WantedBy=multi-user.target

Move the service (again, I recommend mv instead of file explorer) into /etc/systemd/system/ (filepath should be /etc/systemd/system/bluetooth-reconnect.service), and enable it and start it:

sudo systemctl enable bluetooth-reconnect.service && sudo systemctl start bluetooth-reconnect.service

And you should be good to go. At least assuming your distro doesn’t have some specific quirk, which I wouldn’t be able to help you with unless I knew what distro you run. Granted, this is my adaptation of what I saw in the linked forum and my own experience with services, I haven’t actually tested this. But even if it has an issue, this will get you 90% of the way there, and there’s a good chance it just works if the forum answers work for your distro.

Also, the method in the forum post you linked is Ubuntu specific, and there’s a good chance that directory has changed since 2015, when it was last answered. You will not break your system by using my method. If it doesn’t work, you can view the logs using sudo systemctl status bluetooth-reconnect.service and you can stop and disable the service like so:

sudo systemctl stop bluetooth-reconnect.service && sudo systemctl disable bluetooth-reconnect.service

The script only really does observation, it doesn’t try to make any writes to the system except for your Bluetooth command, which would not cause system breakage. It just listens for signals on the dbus that GNOME sends when the screen is locked/unlocked. If you wanted to completely “uninstall”, you’d rm the files from their respective locations afterwards. So please don’t do a clean install to test this, that’s entirely unnecessary and far too much work for such a simple task. Even if there’s a syntax error, the service will just exit, which is no problem.