Plot problems: vertical lines and letters

ListPlot

coords = {{1, 2}, {3, 2}, {3, 1}};

labeledcoords = MapThread[Labeled, 
    {coords, {"A", "B", "C"}, {Above, Above, Below}}];

ListLinePlot[labeledcoords, 
  PlotRange -> {{0, 4}, {0, 3}}, 
  AxesLabel -> {T[k], P[Subscript[p, a]]}, 
  AxesStyle -> Arrowheads[.03],
  Ticks -> {Thread[{{1, 3}, Subscript[T, #] & /@ {1, 2}}], 
      Thread[{{1, 2}, Subscript[P, #] & /@ {1, 2}}]}]

To get a hand-drawn look use Simon Woods' xkcdConvert:

 xkcdConvert @ %

Graphics

Graphics[{Thick, Line[Partition[coords, 2, 1]], 
    MapThread[Text, {Style[#, 14] & /@ {"A", "B", "C"}, 
        coords , {{0, -1}, {0, -1}, {0, 1}}}]}, 
  PlotRange -> {{0, 4}, {0, 3}}, Axes -> True, AxesOrigin -> {0, 0}, 
  AxesLabel -> {T[k], P[Subscript[p, a]]}, 
  AxesStyle -> Arrowheads[.03], 
  Ticks -> {Thread[{{1, 3}, Subscript[T, #] & /@ {1, 2}}], 
      Thread[{{1, 2}, Subscript[P, #] & /@ {1, 2}}]}]

ParametricPlot

ClearAll[f]
f[x_?NumericQ] := Piecewise[{ {2, 1 <= x < 3}, {1, x == 3}}, Undefined]

ParametricPlot[{x, f[x]}, {x, 1, 3}, 
  PlotRange -> {{0, 4}, {0, 3}}, AxesOrigin -> {0, 0},
  AxesLabel -> {T[k], P[Subscript[p, a]]}, 
  AxesStyle -> Arrowheads[.03], PlotStyle -> Black,
  Ticks -> {Thread[{{1, 3}, Subscript[T, #] & /@ {1, 2}}], 
      Thread[{{1, 2}, Subscript[P, #] & /@ {1, 2}}]},
  Epilog -> MapThread[Text, 
   {Style[#, 14] & /@ {"A", "B", "C"}, coords, {{0, -1}, {0, -1}, {0, 1}}}]]

Try this:

Show[{
  Graphics[{Line[{{1, 2}, {3, 2}}], Line[{{3, 2}, {3, 1}}]}, 
   PlotRange -> {{-0.10, 4}, {-0.1, 2.5}}, Axes -> True, 
   AxesStyle -> Arrowheads[0.03], 
   Ticks -> {{{1, 
       Style["\!\(\*SubscriptBox[\(T\), \(0\)]\)", Italic, 16, 
        Black]}, {3, 
       Style["\!\(\*SubscriptBox[\(T\), \(1\)]\)", Italic, 16, 
        Black]}}, {{1, 
       Style["\!\(\*SubscriptBox[\(P\), \(1\)]\)", Italic, 16, 
        Black]}, {2, 
       Style["\!\(\*SubscriptBox[\(P\), \(2\)]\)", Italic, 16, 
        Black]}}}, 
   AxesLabel -> {Style["T(K)", 16, Italic, Black], 
     Style["P(Pa)", 16, Italic, Black]}],

  Graphics[{Text[Style["A", Black, 16], {1, 2.15}], 
    Text[Style["B", Black, 16], {3, 2.15}], 
    Text[Style["C", Black, 16], {3.15, 1}]}]

  }]

with the effect

To my taste this is more straightforward than to draw it by equations.

However, you definitely can draw these lined using standard Plot-family functions. Consider, for example, this:

Show[{
  ParametricPlot[{1, y}, {y, 0.5, 2}, 
   PlotRange -> {{0, 2.5}, {0, 2.5}}],
  ParametricPlot[{x, 1}, {x, 0.5, 2}]
  }]

yielding

Have fun!