My ceilings are flat and I haven't had any reason to mop them, so sorry, I've nothing to bring to the table.

What is up there? Are you trying to clean grease off a wood tongue-in-groove ceiling? How often do you need to do this?

I once had one of those Microwave recirculators. That was ugly. The solution was to vent the greasy air to the outdoors.

But I did need to clean grease off the cabinets and walls and ceilings overhead. I had to build a scaffold to wash it down.

I imagine your best bet is a sponge mop. And one of those ettore window washing extension poles. If it's too high up for that, seek professional help.

Good luck!