The Wallbox itself costs between 500 and 1,500 euros depending on the model, and installation by an electrician adds another 1,000 to 2,500 euros - heavily dependent on how far away your meter is and whether the existing electrical system is sufficient. With your landlord, you definitely need written permission, that's not optional, and honestly the bigger hurdle than the costs themselves. Good news: there's KfW funding (programs 441/442) with up to 900 euros in subsidies per Wallbox, and Baden-Württemberg sometimes adds a bit more on top - check out the L-Bank.
Before you get a quote from the electrician, you should clarify: what's your house connection rated at (usually 3x16A or 3x32A), do you need a new line to the parking space, and can the meter even handle a higher load or does the grid operator need to get involved. If your landlord agrees, he usually demands that you split the costs or that the box gets removed when you move out - that should be settled beforehand.