If you purchased property after January 1, you may not receive a tax bill in your name until the following year depending on when you purchased the property. Your name may appear as a c/o or not at all.
However, depending on your closing statement, you may be required to pay the tax bill in the "previous owner’s name". This should be determined at the time of closing and the adjustment should be reflected in your closing costs. Tax pro-rations are typically based upon the most recent tax bill available.