Does anyone know where/how I can upgrade the OS on an ASR 10R unit? Is it via a floppy disk or is it hardware?

The OS is on floppy disk. What you need to do is go to http://www.emu.com and hunt through all their downloads... They have it there. You'll also have to download a program to write it onto floppy disk called Giebler. They'll have a link to it right there on the same page as the OS.

The latest OS version is 3.53, btw. Make sure that's not the one you already have, otherwise you'll have gone through all this for nothing.