Using (10.087 +/- 0.044) x 10(9) J/psi events collected with the BESIII detector at the BEPCII storage ring, the processes Lambda p -> Lambda p and (Lambda) over barp -> (Lambda) over barp are studied, where the Lambda/(Lambda) over bar baryons are produced in the process J/psi -> Lambda(Lambda) over bar and the protons are the hydrogen nuclei in the cooling oil of the beam pipe. Clear signals are observed for the two reactions. The cross sections in -0.9 <= cos theta(Lambda/(Lambda) over bar) <= 0.9 are measured to be sigma(Lambda p -> Lambda p) = (12.2 +/- 1.6(stat) +/- 1.1(syst)) and sigma((Lambda) over barp -> (Lambda) over barp) = (17.5 +/- 2.1(stat) +/- 1.6(syst)) mb at the Lambda/(Lambda) over bar momentum of 1.074 GeV/c within a range of +/- 0.017 GeV/c, where the theta(Lambda/(Lambda) over bar) are the scattering angles of the Lambda/(Lambda) over bar in the Lambda p/(Lambda) over barp rest frames. Furthermore, the differential cross sections of the two reactions are also measured, where there is a slight tendency of forward scattering for Lambda p -> Lambda p, and a strong forward peak for (Lambda) over barp -> (Lambda) over barp. We present an approach to extract the total elastic cross sections by extrapolation. The study of (Lambda) over barp -> (Lambda) over barp represents the first study of antihyperon-nucleon scattering, and these new measurements will serve as important inputs for the theoretical understanding of the (anti)hyperon-nucleon interaction.