State tax filing

I am in a similar situation to the original person. I live in Texas, but was hired by a New York based  company to work remotely. Working remotely was offered in the job description, and was more beneficial to the company than my self.  I will have to travel to NY once a month or so for a week at a time, i understand i will have to pay taxes for those days. I meet many of the conditions from the link you posted for "Office of Tax Policy Analysis Technical Services Division". The closest office is ~1,200 miles away. 

Do I need to fill out my W4 for NY in  a particular way?
Do I need to pay the taxes through out the year and then complete my taxes in a particular way to receive that paid money back?