Mud for patch 4 inch drywall holes
I'm having a hard time articulating this without a visual aid, so bear with me.

You can eliminate the need for paper tape by leaving one side of the paper on the drywall patch after you score / snap it. So instead of cutting the other side of the paper after the score / snap, you peel it back an inch or two on all sides, and cut the paper so that you'll have a sheetrock patch with a built-in "tape" to overlap the seam. This youtube video demonstrates the idea:

https://www.youtube.com/watch?v=qOSzbX92ttM

This is easier for patches with straight lines that you score easily, but I think with some finesse you could do the same thing for a circular patch, especially if you have the right sized hole saw. 

What the other guy said would work great too - and I 100% agree that any texturing should be done afterwards. You may even want to throw on a layer of primer before you texture to make it hold better. 

Also, you can get canned spray-on texture to use on a patch job like this, which should make the job simpler if you don't have all the other equipment. They come in a few varieties so you can match the existing texture.