I think he meant he would save $2,000 once by not having to get a CELTA, not that he would save $2,000 every month.
OP: I have no idea about whether you could rent a place monthly, though in Taidong, things would be far cheaper generally. Renting any place (monthly or not) initially without an ARC might be difficult. I think you’re going to have to pay for temporary accommodation. Maybe you could make a deal with a cheap hotel to rent a room for a month for a fairly low price (try 10,000-15,000NT/month). Otherwise, if you can get shared accommodation, that would be good though how you would do that is anyone’s guess.
As has already been mentioned, getting work in Taidong is going to be very difficult. It’s simply neither big nor rich. The good jobs for Taiwanese people are in other parts of the country, and so the teaching jobs are also. Taidong is cool (I like the whole east coast immensely), but there just isn’t much work. It’s also a long way from the bigger cities with nightlife, etc.
Also, people in Taidong probably don’t speak much English at all. In fact, plenty probably don’t speak Chinese that well either. According to my wife, Taiwanese is far more common.
As for preparation? Money is good. Being able to speak some Chinese (or Taiwanese) would be good. Having the right documents would be good. Other than that, I don’t know. You’ll largely have to figure it out as you go.