This is very hard to tell what will exactly happen. The best possible solution could be to first let your customer know about the situation and what is required. You should not take the decision of updating stuffs on a live website of your own. You should let your customer know about the downside of such an action since he did not update the theme for a long time as you have mentioned.
Next, you need a development platform for this to implement and test everything (e.g. demo.brechtgebaeudereinigung.de), make all adjustments that may come up and once you and your customer is happy with everything, move things to production environment.
However, the theme developer knows about it the best and they can give you the best solution, but the minimum you should do is taking backup of the site and database before going for an update. Since you did not mention exactly how long theme was not updated it is hard to tell how much “breaking changes” you will end up with after the update.
So, the first and foremost thing is to take backup of everything before going for the update, then create a development environment and test things there.
Hope this gives you some light about how to proceed.
