Is construction power worth it because it is cheaper? Experiences?

The expensive part of construction site electricity is not the electricity itself but the box from the electrician, because you rent it.
For us, I think it was 3 euros per day.

The electricity is of course also correspondingly expensive depending on the amount, but the electricity tariff does not care whether it goes through a construction site meter or a regular meter.

For us, the "hardware" also comes from the construction company and is not charged separately. Therefore, I don't care which meter the electricity runs through.

Do I have to rent the box from the electrician or can I also use the existing one? Are there any differences?

The distributor and meter were installed by the previous owner to have electricity on the property (in the garage). So, could I also use the existing distributor/meter for the construction power? Then I wouldn’t have to install or rent a construction power box, right?

You have to install your own meter or have it installed by an electrician. The electrician has to report it. At the latest, when the previous owner receives a bill, they will present it to you. You should ask him if you are allowed to use his box.