Power on S3 Mini without working power button
Okay my solution was to buy a cheap Samsung JIG from eBay (~2.5€). You can also build one yourself, but I did not have any resistors laying around at the time, so I just chose to buy on already made.

I plugged it into the mobile and after just a few seconds it turned on into download mode.

I then removed the jig, and connected my samsung s3 mini to my computer. I had to install the correct drivers and started odin. I did not find a way to simply boot my device, but I was able to create an emptpy .tar file to flash. I chose this file via the "BL" button in Odin. I made sure "Auto Reboot" was enabled and hit Start. It worked like a charm. Because the tar file I provided was empty it did not actually flash anything, but the device turned on.

I can now turn on the device whenever I need, using the jig and  my pc.

One other problem I ran into is: How do I turn the screen off, having no working power button. The Solution: Apps. There are a number of apps out there which enable you to turn off the screen via many different methods. Choose one that fits you best :)