Thanks for the responses.
As I said, importing the PDF file garbled up totally and .ai didn't work either. I tried that, because it obviously would be the best solution (but wouldn't solve my problem anyway, because I think the agency didn't include a bleed anyway and the size and format was for the US anyway, so I think it was best to adapt it for my market anyway).
Now to the actual problem: I still have some credit with this printer from a refund so I would like to go with them. And unfortunately they only accept .cdr or .jpeg for now (yeah, I know...).
The information about size the give for the 5x9cm business cards:
Final size: 5,08×8,8cm
File size with bleed: 5,38×9,1 cm
And in the general instructions they as for 3,5mm of free space from the borders of the document (1,5mm bleed, 2mm security space).
What I did now was create a file with the size of 5,38x9,1mm and made the security margin 3,5mm and a bleed of 0mm (so it includes the bleed and the security margin) and now Scribus will export the whole file.
I am not quite sure why Scribus doesn't export the bleed if there is one. As far as I understand the printer needs the file with the bleed, so the background color or image is over the whole document and there are no white borders.
Thanks anyway for anyone who helped with an answer.