We study the ground-state phase diagram of two-component fermions loaded in a ladderlike lattice at half filling in the presence of spin-orbit coupling. For repulsive fermions with unidirectional spin-orbit coupling along the legs we identify a N\'eel state which is separated from rung-singlet and ferromagnetic states by Ising phase transition lines. These lines cross for maximal spin-orbit coupling and a direct Gaussian phase transition between rung-singlet and ferro phases is realized. For the case of Rashba-like spin-orbit coupling, besides the rung-singlet phases two distinct striped ferromagnetic phases are formed. In the case of attractive fermions with spin-orbit coupling at half filling for decoupled chains we identify a dimerized state that separates a singlet superconductor and a ferromagnetic state.