due to mainboard failure I had to replace the hardware. I took the old disk drive and connected it into the new hardware. The system starts, Windows is coming up and all of a sudden ending in a blue screen.
The last message indicates, that there problems accessing the file system. I assume, this is because of the missing disk drivers chipset drivers.

Since I can (temporarily) install and access this disk into an other system, how can I add / install additional drivers to the existing Windows installation.

Thank you for your help

Try booting in safe mode (hit F8 repeatedly when BIOS screen is on its way out) and select safe mode.
Sometimes you can add the new drivers if Windows doesn't blow up.

If not dice, take all the INF and SYS files for your new hardware from the driver folders, and copy them to the respective folders (c:\<WINROOT>\INF and c:\<WINROOT>\system32\drivers) and hope that Windows can find them.

The problem is probly that Windows can't find the driver to access the motherboard chipset. If safe mode fails, copy the INF and SYS files to the folders on the drive, and then reboot using safe mode. Most times, it should find the new hardware.