Installing Remote Server Administration Tools on Windows 10 is a bit challenging at times. At least, it was for me: The installation would start, then hang, then fail with "critical error: 2147549183". Here's how I got it to run.
Language pack. I have a German installation of Windows 10, and Google suggests I need the English language pack. I suspect this info is outdated, but I installed it anyway. Search in the start menu for "Sprache auf diesem Gerät hinzufügen", then add English (United States).
Start the installation up to the point where you accepted the license and everything, and it says "Update <weird ID> is being installed".
The installer now hangs. From C:WindowsLogsDISMdism.log I got the impression it is waiting for some resource which it can't get access to (the log talked about some resource being busy).
Hit the reset button. Initially I had intended to clear whatever locks the installer needs by rebooting, but I couldn't reboot because it hung, so I just hit reset. What I hadn't anticipated then was that...
When rebooting, Windows notices there's still an update in the queue and applies it.
Now you're good to go.