Can I use German flour Type 405 for a recipe that needs self-rising flour?
The keyword here is “self-rising” - which describes a mix of flour and baking powder.

So if you are going to use 405 instead, which would probably be a good—enough choice (especially as it’s “the standard flour” in German supermarkets), you have to add baking powder. As a rule of thumb, use one pouch (assuming you are in Germany, where that’s the standard unit) per 500g of flour. If your recipe calls for self-raising plus baking powder or baking soda, use the extra amount as well.

I personally prefer he German 550 flour as my go-to type, which is closer to what other countries call “all-purpose”, but it may or may not be available at your store. Unless you are making bread and similar goods, 405 will most likely be ok.