Rust is always a problem with steel or cast iron. You will need to remove it before you can repaint the mower.
Rust can be removed using a wire brush. Glass paper, sanding blocks and wire/steel wool are also useful. These all create dust so work in a well ventilated area if possible and wear a face mask to prevent breathing it in.
A wire brush wheel in a power hand drill can cut a lot of the effort out but can create even more dust. As with all power tools, take care and wear appropriate protection.
If you have access to shot blasting equipment this can make cleaning the components a lot easier. Shot blasting can clean the metal right back to a bare and clean surface. In reality it is not necessary to go this far.
Special chemicals can also be used to remove rust but there is no substitute for hard work with a wire brush.
You should be aiming for a surface free of loose rust and paint.
Even the rustiest machines will have some paint deposits on them.
There is a good case to be made for leaving original paint on the mower even if you are going to paint over it. In this way the original paint will be preserved for future inspection by historians, industrial archaeologists and enthusiasts may be able to use as yet unforeseen techniques to learn about the materials and skills of the past.
If you do decide to remove the paint, scraping is one of the simplest and best techniques, particularly for small areas. Scraping is also useful for larger areas but additional methods may be required. Paint stripper is useful. The water soluble, semi-gelatinous types are the best as they are easy to use and can be washed off afterwards.
Brush the stripper over the parts and leave for 15-30 minutes, using the time to work on other bits.
Once the paint is softened it can be removed more easily by scraping or wire brushing. A second or even third application is sometimes required.
Wipe cleaned parts with a rag soaked in white spirit to remove any stripper residues and leave to dry.
Remove remaining rust as previously described.
Some parts of the mower may be covered in oil or grease deposits. These need to be removed before the parts can be painted.
Heavy deposits can be scraped or wire brushed. After this, use special degreasing fluid to remove the remainder. Water washable is best.
For heavy deposits, soak parts in diesel oil followed by a final clean using degreaser.
When the parts are clean, remove any paint or rust as previously described.
If the parts will not require final painting (chains for example) store them safely until the mower is reassembled. A light coating of oil will help prevent new rust forming and will stop the components seizing while you work on the rest of the mower.
Rust forms quickly on bare iron and steel. When the parts are clean give them a coat of priming paint as soon as possible.
You are now ready to paint the mower.