Yes!! My improvised PSU works like a charm, and my 5Big is up and running again! And now I see, that the firmware-update was actually successful
I thank all of you for your advice and ideas - I'll never again underestimate the power(lessness) of a power supply!
andreus, just a quick word about your capacitor.
Some years ago I repaired 2 or 3 pc-mainboards myself by changing broken capacitors with fresh ones. I used a cheap soldering iron, and I didn't care too much about heating up the board .. even if pc-mainboards are 4 to 6 layered, in the end everything worked fine for another couple of years
So I don't think you could do so much wrong. Changing a capacitor is quite straight forward .. heat up one of its legs, and pul the capacitor out the other side .. if the first leg is loose, heat up the other one, and pull the thing out fully. Normally most of the solder goes out with the legs, so you are left with two empty holes. Push the replacement capacitor (take care, that you mount it right! The black or white line with the arrows needs to be on the same side as before!) into these holes, and put some solder to the legs. voilá - repaired
I recommend a capacitor with the same µF but a little more volts .. that makes it more durable in the future.