China Town, Little Italy, SoHo, Greenwich Village in NYC
 

Is it feasible to try to hit all these places in one day?

When walking through these areas, what is the best order to do them in?

Depends on where you want to end, but I would do:

Chinatown
Little Italy
Soho
Greenwich Village

Yes, depending what you want to do in each place. If you're a shopper, you could spend the whole day in SoHo. I suggest being in Chinatown around meal time.

Yes it's feasible.

Chinatown can be quite frantic with the street grocers and vendors, especially if you come from a smaller city it could be a minor culture shock, so I would recommend hitting it first, instead of when you are exhausted from walking all day. Agree with Barry about hitting it at lunch, then walk over little Italy, then west to soho, and ending in the village for a tea/coffee break. Note that Greenwich Village is north of soho, whereas the West Village is west of Soho, different neighborhoods but both nice for strolling and browsing.

Eat lunch in Chinatown then walk across Canal St to Little Italy for a cannoli for dessert.