Should I clean (how?) old drywall before applying mud?
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.

https://www.homedepot.ca/product/t-s-p--trisodium-phosphate-powder-400-g/1000130976

Paint with primer.

Then tape and mud.