need some advice how to repair the chair
starting the project
Deciding whether to repair myself or take it somewhere to have it repaired
Welcome to the Bunnings Workshop community @PetrinaG. It's sensational to have you join us and many thanks for your question on rattan chairs.
If this was only slight damage to the armrest, you might have been able to sand the surface back to remove the loose fibres and then refinish it. However, from your photos, it looks like in several areas a significant portion of the cane has been worn away, so simply sanding it back might not restore the appearance. It could still be worth sanding the damaged areas with 180 grit sand paper to see how they come up.
Having the chairs professionally repaired is certainly an option if you'd like to preserve them, but if you're keen to tackle the repair yourself, one approach would be to rebuild the missing thickness first. Because there appears to be quite a bit of cane missing from the armrests, you could use a packing material such as timber strips to restore the shape, then wrap the entire armrest with split cane to conceal the repaired section and give it a more uniform finish.
Let me mention @DIYhackjob, @annieerbs, @ang2, and @becster, to see if they have any thoughts. You might also like to check out some of the previous discussions on repairing rattan furniture via the search function at the top of the page. Some of those include:
Please let me know if you have any questions.
Mitchell