You use a hot TSP (tri sodium phosphate) solution on a mop. The mop is kinda important at least for the last step, normal wiping may produce streaks. You may still end up with blotchy areas BUT you had to do this anyway to get the grease off to ready it for painting if that is still needed.
Important - wear eye protection and cover skin. It is not horrible to get it on skin but will pull the oils out so should not be a long exposure.
If it is plaster and you get it wet and mop it you will ruin the finish for sure.
If it were me, I would hose it down with brake cleaner. It will strip the grease off and perhaps not damage the plaster finish because of the lack of water? I've never sprayed brake cleaner on plaster but if you scrub it with anything it will destroy the finish. I think it's worth a shot if you really want to get the grease off while preserving the finish. Just put a pan underneath.
There's an option 3: stick something over the area and take it down to wash / replace regularly. If you use something reasonably stylish like peel and stick back splash, it should look ok.
If you're at the point where you need to tarp the kitchen ceiling, other cooking arrangements should be considered.
EDIT: My idea is your kitchen is old enough to drive, it's time to renovate.