1. Expect to pay $1000-1500 for an apartment rental
2. Hard to say how far it will be from the school. Get the address of the school and look for apartments within walking distance. This may be difficult, so you may want to look into renting a car for the month.
3. Little or no public transportation aside from Metro North's Commuter Railroad, with Brewster proper stops at Southeast and Brewster. These trains make stops in Putnam, Westchester, and Bronx counties en route to Grand Central Terminal in Manhattan.
4. Can't really help you on the tax question, but since you are only staying for a month, it shouldn't be too much. Ask the school that you will be teaching at. They should be able to give you an idea.
Best of luck.