How should I patch a popcorn ceiling?
No - It will never match the original pattern. But we had similar issues in many apartments where we couldn't just leave a bold spot. So we had to re do that "area"  area being keyword here. 

Afterwards apply some undercoats to the area that was redone (we did ours with a concrete mix but some have silicone or other types)

Only you will see exactly where the repair work has been done but trust me. Nobody else will even notice the difference because they would really have to know they are looking for something.

Basically applying a similar patter and repainting the entire area will take the obvious repair work out of eye sight. Just don't tell any body because then everybody will see it.

(That is why I hate doing my own DIY work because I always find something somebody else messed up or the tiny shortcut I took that nobody else ever will even know about)