When you say original xbox people may think you mean one of actual original big black behemoths, not a stock 360.
Anyway simple option is put the profile + saves on a USB (later dashboards added the option to do this, later ones still made it so you don't even need something like usbxtaf any more) or 360 memory card and transfer that over. If however you have hacked games and other things you might not want on that profile then that might not be ideal if you don't want to be banned from live (I don't know how hot MS are on all that these days -- they care a lot about hacked in achievements for whatever reason and cheat modded saves but other things are more hazy).
There are more hackery ways like take the profile off the stock 360, take the one off the RGH machine, stick them both on a PC and then take the stuff from the RGH machine and inject the saves (not sure what will go for minecraft and locking it to profiles) and vice versa. Not sure what the kids are using for this sort of thing these days, though
https://www.wemod.com/horizon will get it done.