Yes on the whole thing. If you don't paint the whole thing, the colors will not match up. If some of the existing paint is still very firmly adhering to the metal and doesn't want to come off with a wire brushing, it should be OK to paint over it after a good sanding/scuffing up.
It might be easier to brush or roll the paint than to spray it; that's somewhat of a personal preference. (Rustoleum and similar paints do roll pretty nicely with a foam roller, possibly with a touch of thinning. With a little bit of care it can give a nice smooth finish that compares very well with spraying.)