No, you can't get the resident rate right away.
Whether or not you can get the in-state rate at all depends on if you are considered a dependent or independent for financial aid and tuition purposes. After a certain age (I believe 25), or if you have dependents (i.e. children), you are considered independent. Also, if you've already received a bachelor's degree, you're considered independent.
If you are considered a dependent, your residency is determined by that of your parents, and therefore, you can't gain residency status if classified this way.
If you are considered independent, most likely (depending on the school), you can have your residency status reconsidered 12 months after demonstrated proof of established residency. This means you have to show your residence is off-campus/non-student housing and in New York state, also you have to show other proof of residency (voting registration is in no other state but NY, driver's license is NY, your car is registered and insured in NY, other bills, etc., are tied to NY, and nothing tied to any other state). It's a lot of work (I know because I did it and succeeded), but it can be done, but only after 12 months of continuous established residency.