TRENTON — Christopher J. Christie took office as governor of New Jersey two months ago vowing a sharp change of direction for a state battered by the recession and choked by its tax burden. On Tuesday, he made clear that what he had in mind was a U-turn. Upending the priorities of his Democratic predecessors, Governor Christie unveiled a budget that would hit the poor, elderly, schoolchildren, college students and inner-city residents hardest, while largely sparing the wealthy and businesses. The plan proposed by Mr. Christie, the first Republican elected New Jersey governor in 12 years, relies almost exclusively on spending cuts, and stands as a stark example of how a Republican committed to not raising taxes grapples with a state budget in these still-challenging economic times.