Hi Falk, Berlin is great! And the weather is super nice these days :)
Regarding your question: you should get the car documents, the driver's license, and car insurance valid in Russia. The insurance might be available at the border, but this is not 100% sure, so I'd double check once you're in Ulan-Bataar — maybe some agency there can provide it for you. Also, yes, you will need a document from your friend confirming that he lets you use his car to go to Russia. In Russian it's called "Generalnaya Doverennost'" or Генеральная Доверенность. This will prove to be a bit more complicated, but if I were you, I'd go to a notary office in Mongolia and ask them to provide this paper to you and to explain where you need it. You need to find a good notary who has experience with these cases, so they do everything right. Then you just get a Russian translation, as well as the translation of the car's technical documents and certificate into Russian (you could do all this at once). And then, yes, you can travel to Russia. Get a membership from some car association like ADAC just in case and ask if they have any partners in Siberia.