what kind of drywall texture is this and type of brush to use to get this texture
I think that's a crows foot pattern made with a texture brush that is stomped/slapped versus the drywall. I think this brush is overkill for a patch job, but the texture it creates seems to match pretty closely with your picture. 
https://www.all-wall.com/Categories/Drywall-Texture-Brushes/Crowsfoot-Drywall-Texture-Brush.html

You may be able to purchase the brush at lowes/home depot but I've found that drywall texture tools availability is hit or miss depending on the location.

I'm not sure if you are familiar with using stomp brushes, but I think there's a couple of videos on youtube that do a pretty good job explaining. 

I would recommend mixing up the mud and practicing on a piece of cardboard/scrap drywall until you can consistently get a similar pattern. 

I just did a lot of ceiling patching, and I ended up having to add a good deal more water to the mud than expected to get the pattern to match (or as close as I could come).