A plot that looks like the one in the picture posted in OP:
Plot[{1, 3, 5}, {x, 0, 35}, ImageSize -> Medium, Filling -> Axis,
PlotRange -> {All, {0, 6}}, ScalingFunctions -> {None, "Reverse"},
AxesLabel -> {"T[°C]", z[mm]}]

1. If it's not essential to have axes labels positioned at the ends, simplest approach is using Frame
and Charting`ScaledTicks[{Identity, Identity}]
to create horizontal axis ticks:
Plot[{1, 3, 5}, {x, 0, 35},
Filling -> Axis, PlotRange -> {All, {0, 6}},
ScalingFunctions -> {None, "Reverse"},
FrameLabel -> {{z[mm], None}, {None, T[Row[{Degree , "C"}]]}},
Frame -> {{True, False}, {False, True}},
FrameTicks -> {{Automatic, None},
{Automatic, Charting`ScaledTicks[{Identity, Identity}]}}]

Use Charting`ScaledTicks[{Identity, Identity}][0, 35, {8, 6}]
to get

2. Alternative approach with more detailed control over positioning and styling of ticks and labels is to use AxisObject
:
Create a version without the horizontal axis:
plot = Plot[{1, 3, 5}, {x, 0, 35}, ImageSize -> Medium,
Filling -> Axis, Axes -> {False, True}, PlotRange -> {All, {0, 6}},
ScalingFunctions -> {None, "Reverse"}, AxesLabel -> {None, z[mm]},
Ticks -> {None, Automatic}]

and use AxisObject
to construct an axis with desired features:
xaxis = Graphics @ AxisObject[Line[{{0, 0}, {35, 0}}], {0, 35},
AxisLabel -> " T[°C]",
TickPositions -> {{Range[5, 35, 5]}, {Range[35]}},
AxisStyle -> Black,
TicksStyle -> {Black, Thickness -> Absolute[0.5]},
TickLabelPositioning -> "Tip",
TickLengths -> {.15, .07}]

Combine plot
and xaxis
using Show
:
Show[plot, xaxis]

Remove the option TickLabelPositioning -> "Tip"
and add the option TickDirection -> Down
in definition of xaxis
above to get

Frame
based solution? $\endgroup$