Can you program any IR remote to a Roku ultra, like you can to an Apple TV?

Sorry, but no. Roku ultra only respond to true Roku IR commands. Which many aftermarkets universal remotes know. 

No Roku remotes are learning remotes.

For using a different remote with a Roku device, all Roku devices except sticks that plug directly into an HDMI port without the need for a cable support IR remotes.  Most universal remotes have codes for Roku devices these days.  The remote that comes with the Ultra is an RF/IR hybrid.  I only seem to be able to find my Voice Remote Pro (rechargeable) right now, so I can't say for sure that the battery powered voice remote with TV controls acts the same (and I don't know which remote the Ultra comes with these days), but if I unplug the power from my Roku device, the remote reverts to IR control.  I could then use that remote to program a learning remote.  If its Wi-Fi direct (RF) connection is established, it no longer transmits anything except TV controls via IR.