If we talk about OEM armrest, this is a complete center console unit. Honda can provide you as well but with some higher cost.
Regarding installation and any sorta damage, None.
Your center console (complete) upto your front cup holders will be removed and the new one with armrest will be installed.
cheaper solutions are universal armrest or plywood crafted locally. Universal will bolt down to your plastic and ruin the plastic finish.
Plywood type (covered with upholstery) will simply rest over you existing console. Also not great looking in my opinion.
Another option is to source out something like this. But particularly for Honda City/Honda Fit 2009 onwards