The repairs that are needed will depend on how bad the rust is. The sooner you treat rust, the better. Over time, you may learn to deal with tiny patches yourself.
For small patches with no structural alteration, the typical treatment is to sand the rust off of the affected area and then repaint. Care should be taken to make the new paint match the old. It’s worth considering repainting the entire fence if you have multiple rust spots or if an inspection reveals that the paint is chipped in other places.
For more serious problems, especially if your fence or gate is sagging, it may be necessary to remove and replace the affected piece, generally by cutting out defective lengths and welding in the new piece. Hand made wrought iron is now an expensive luxury, but a good repair company will replace it with more modern iron which will look and “act” like the original. Beware cheap repairs that might use, say, aluminum for the replacement (aluminum is commonly used to make fake wrought iron, but is not nearly as strong or dent-proof as the real thing). You will also need a repair company that has the skill to shape the new iron to match the old. This is not a job you can hand over to just anyone.