Thanks for investigating and helping! :) Serial-to-USB-Connector is ordered^^.
Okay, I think I figured something out. I tried some of the "Archived Version" of Armbian ( https://armbian.systemonachip.net/archive/nanopineo3/archive ):
It seems that 21.02.1 (Buster) didn't run either on my Neo3; same behavior (like, Router never get a DHCP-Request).
Armbian 20.11 (Buster) works! With that Image, I could finally use the NanoPi.
* First thing I did: "apt-get update; apt-get dist-upgrade". Everything runs fine, than "reboot"... And nothing works anymore. Router never get a DHCP-Request after Rebooting.
* After that, I flashed my SD-Card again (20.11 Buster), I changed a Package to hold ("apt-mark hold linux-image-current-rockchip64"), and update again. Update runs through, I could reboot the NanoPi and... everything still works!
So, if I connect to the NanoPi using SSH, it shows:
"Welcome to Armbian 21.02.3 Buster with Linux 5.8.17-rockchip64"
Whatever the Problem is, it has something to do with the Package "linux-image-current-rockchip64"...
SD Card is a SanDisk Extreme, 64 GB, U3 A2. I tried another SD-Card (SanDisk Ultra, 16 GB, Class 10), same result.
After my Serial-to-USB-Connector arrived, I'll try to find out what's going on...