I just use a thick nap roller. It should paint just fine.
Consider using a roller.
There are thick nap rollers made just for textured surfaces.
I have painted my textured ceiling. If your texture is the spray on type, be very careful. Paint is like water-- it will take this type of texture off. This is the kind my texture was. Use a brush to trim the edges, and a roller for the main part. You will want to prime first if it has never been primed. The main thing to remember with the type of texture that I am referring to is to roll or brush over it only once and then move on. If you go back over the wet area, you can remove the texture. Even though you will be tempted to add "just a little bit more primer" don't! After the first coat of primer is dry, you can go back over again--still just using one pass. You can test your type of texture in a closet or corner of a less used room by taking a wet cloth and wiping a small area. If your texture comes off, please be careful when priming and painting.
Take a good photo to the paint dept. at your home improvement store and ask for their professional advice on materials and application methods.