What to prepare for automatic irrigation

That's exactly what I thought. A computer like the one in the picture can only control one zone. The larger one can control a maximum of two zones. With that, you won't be able to automatically water your garden.
Therefore, better use a distribution box, control cable (not power), and a proper irrigation controller.
At the spot where you place the controller, you need power, Wi-Fi, and the control cable.

Many underestimate how much water is really needed
Lawns quickly require 25-30 liters / m2 per week during the heat period.
Let's subtract possible hedges, flower beds, and trees; from the 300 m2, a conservatively estimated 200 m2 of lawn area remains.
Just for this alone, 6,000 liters of water are needed per week.... Hedges, trees, and flower beds still need to be added.
The demand is even greater during the establishment phase.

And can I supply a distribution box with water via the faucet? Can you give me an example of a proper irrigation controller? By the way, we do the electrical work with free@Home, something compatible with this would of course be good

Of course, you could connect a distributor/valve box to the faucet. I have seen that Gardena also has a distributor for 6 zones including control. If you want to keep it very simple, it’s worth considering. Also very cheap (Gardena water distributor automatic). For valve box + computer, I would recommend Hunter Hydrawise as already mentioned. Free@Home is not my area. It’s too limited and closed for me. Instead, I (still) rely on KNX. There might also already be free@Home compatible computers (Hydrawise is not one). Theoretically, you can also control the channels directly via free@Home: setting timers, creating visualization... just requires knowledge and time.

If you have Free@Home, in my opinion you don't need the computer, provided you have enough capacity on your actuators. Just connect all the valves to an actuator and you can set them up and control them via Free@Home.

Please note, however, that you will then need different valves.