Joseph Lhota, the M.T.A. chairman, will step down on Friday and is expected run for mayor as a Republican. A law prevents him from entering the race while holding the chairmanship. He's received widespread praise for his response to Superstorm Sandy. But the agency will adopt its new budget on Wednesday, and is expected to vote on approving a slate of fare and toll increases. It is likely he'd face primary challengers as well.