Sorry, but my only comment is 'lots of luck'! Maybe someone has the magic answer, but texture is very dependent on the installer. Trying to replicate it can be really tough.
I would generally call that "orange peel" texture, and there are spray-textures available in that. But texture is very much a one-of-a-kind thing when done, and matching is an art form. You might look for a skilled drywall repair person.