Are you asking how to paint a wall covered with lattice, paint faux lattice on an existing wall or how to paint lattice in general?
Generally I paint lattice with a paint sprayer.
Exterior lattice can be painted w/exterior house paint. You can apply the paint using a brush or roller but the easiest, fastest option is to spray paint it on. Krylon and Rustoleum offer a vareity of paints in simple spray cans. Be sure to cover whatever areas you don't want the paint to get on. If the lattice is already in place, you can likely slide sheets of newspaper behind it. Good Luck!