I have screwed a new piece of drywall. Now I need to try to apply mud and finally paint. Should I clean, wash the old painted drywall first? If yes, how? It does not stand out in this picture but the old drywall is very dirty.
Clean with TSP. You can use a sponge for the tsp. I've used a big bucket and a bee mop. You just need to be able apply some force while cleaning. In your case that is a small area, a sponge is good.
Paint with primer.
Then tape and mud.