DirCond = DirichletCondition[\[Phi][r, z] == 0., r == rmax];
NeumCond = NeumannValue[0., r == 0] + NeumannValue[0., r == rmax] + NeumannValue[0., z == zmin];

solution = NDSolveValue[{D[r*D[\[Phi][r, z], r], r]/r + D[\[Phi][r, z], {z, 2}] == NeumCond, DirCond}, \[Phi], {r, z} \[Element] mesh, "ExtrapolationHandler" -> {Automatic, "WarningMessage" -> False}]

DirCond = DirichletCondition[ϕ[r, z] == 0., r == rmax];
NeumCond = NeumannValue[0., r == 0] + NeumannValue[0., r == rmax] + NeumannValue[0., z == zmin];

solution = NDSolveValue[{D[r*D[ϕ[r, z], r], r]/r + D[ϕ[r, z], {z, 2}] == NeumCond, DirCond}, ϕ, {r, z} ∈ mesh, "ExtrapolationHandler" -> {Automatic, "WarningMessage" -> False}]

With the following output for t=0.4 (top) and t=0.7 (bottom):

With the following output for t=0 (top), t=0.4 (middle) and t=0.7 (bottom):