Okay, got it to work somehow...
I already installed the STM-32 core with the boards manager and selected the STM-32 board.
First nothing happened when flashing using ST-Link. No error, arduino said that the upload was successful, but nothing happened. I copied the ST-Link command to the windows command line instead, and this time I got an error (mfc100.dll missing). I had to install Visual C++ redistributable package 2010 x86 (on my x64 machine) for ST-Link to actually work.
Now using the "Nucleo-64" board and variant "Nucleo F030R8" the program can actually be flashed via ST-Link, but the pin layout is messed up (D3 in code is D13 on the board).
No idea how to write my own / correct F303K8 variant ... But looks like it might be possible.