I've only really studied Chinese and Korean Medicine in terms of herbs therefore it will most likely differ to Western uses. Anyhoo0
Mugwort comes in several forms, dried being most common, if you go to asian groceries you might be lucky enough to find a fresh bunch, or most asian stores will have it in powder form (handy to keep a bag of mugwort powder)
Mugwort is often used in cooking all around Asia, people believe it promotes good health in general so for important occasions, eg. birthdays or New Years or weddings etc. mugwort is used in a lot of cooking or the powder is used to decorate and give flavour to sweets to give as presents symbolic for good health in the year/s to come. Mugwort powder has a deep green colour and mugwort in general has a sort of bitter and medicinal taste to it (I don't know why its used in things like mochi but its tradition I guess)
Fresh ground up mugwort can be applied topically to the skin (if its too dry add a tsp of Almond oil). Mugwort as antibacterial as well as local anaesthetic factors, relieves burns and rashes as well as arthritis and muscle pains. Fresh sprig can also be chewed to relieve toothaches and headaches.
Mugwort powder can be made into herbal tea, its good for relaxation, relieving mild anxiety and stress (great so great for exam periods). Drink a warm cup of it before bed, sweetened with something obviously. The tea is also used to regulate menstrual cycles, so if you're menstrual cycle is irregular or extremely painful or long excessive bleeding, ginger tea with a tea spoon of Mugwort powder might help. WARNING: drinking excessive amounts of mugwort might induce miscarriages.
A chilled version of the tea is also given to people who have acid reflux or prone to stomach and other gastric ulcers, as it creates a nice or thickens the protective mucous coating.
Dried Mugwort is often burnt as incenses, its believed that the smoke of the mugwort is good for your lungs, so if you have a chesty dry coughs breathing the mugwort incense smoke or the steam from a warm mug of mugwort tea is said to help. WARNING: Mugwort causes mucous secretion so people with medical conditions or already have excessive mucous should avoid.
Now to answer your question :) I reckon burning it as incense will do best, maybe combine it with other herbs of your choice (in my opinion mugwort is not the best smelling incense)