Solution with iterative method in a loop
U[0][x_, y_] := Sin[Pi x] Exp[- Pi y];Do[U[i] = NDSolveValue[{f = Interpolation[ Table[{x, NIntegrate[U[i - 1][x, y], {y, 1/2, 2}, AccuracyGoal -> 5, PrecisionGoal -> 4]}, {x, 0, 1, .1}]]; Laplacian[u[x, y], {x, y}] == 0, u[0, y] == 0, u[1, y] == 0, u[x, 2] == Sin[Pi x], u[x, 0] == 1/2 f[x]}, u, {x, 0, 1}, {y, 0, 2}];, {i, 1, 10}]Visualization
{Plot[Evaluate[Table[U[i][x, 0], {i, 1, n}]], {x, 0, 1}, PlotLegends -> Automatic], Plot3D[U[n][x, y], {x, 0, 1}, {y, 0, 2}, ColorFunction -> "Rainbow", PlotTheme -> "Marketing", PlotRange -> All, MeshStyle -> White]} 