You can use PlotRange -> {Automatic, {1, All}}.

An example:

SeedRandom[1]
data = RandomReal[100, {100, 2}];
data = Append[data, {{500, 50}, {50, 500}}];

Row[{ListPlot[data,  ImageSize -> 300, 
    PlotLabel -> "PlotRange -> Automatic"], 
  ListPlot[data,  ImageSize -> 300, 
    PlotRange -> All, PlotLabel -> "PlotRange->All"]
  ListPlot[data,  ImageSize -> 300, 
    PlotRange -> {Automatic, {70, All}}, 
    PlotLabel -> "PlotRange -> {Automatic, {70, All}}"  ]}, 
 Spacer[20]]

You can use PlotRange -> {Automatic, {1, All}}.

An example:

SeedRandom[1]
data = RandomReal[100, {100, 2}];
data = Append[data, {{500, 50}, {50, 500}}];

Row[{ListPlot[data,  ImageSize -> 300, 
    PlotLabel -> "PlotRange -> Automatic"], 
  ListPlot[data,  ImageSize -> 300, 
    PlotRange -> All, PlotLabel -> "PlotRange->All"]
  ListPlot[data,  ImageSize -> 300, 
    PlotRange -> {Automatic, {70, All}}, 
    PlotLabel -> "PlotRange -> {Automatic, {70, All}}"  ]}, 
 Spacer[20]]

This works in both directions. You can use PlotRange -> {Automatic, {All, 50}} to have the vertical plot range go from the minimum of data to 50:

ListPlot[data,  ImageSize -> 300, 
  PlotRange -> {Automatic, {All, 50}}, 
  PlotLabel->"PlotRange -> {Automatic, {All, 50}}"  ]

You can use PlotRange -> {Automatic, {1, All}}.

An example:

SeedRandom[1]
data = RandomReal[100, {100, 2}];

Row[{ListPlot[data,  ImageSize -> 300], 
  ListPlot[data, PlotRange -> {Automatic, {50, All}},  ImageSize -> 300]}, 
  Spacer[20]]

You can use PlotRange -> {Automatic, {1, All}}.

An example:

SeedRandom[1]
data = RandomReal[100, {100, 2}];
data = Append[data, {{500, 50}, {50, 500}}];

Row[{ListPlot[data,  ImageSize -> 300, 
    PlotLabel -> "PlotRange -> Automatic"], 
  ListPlot[data,  ImageSize -> 300, 
    PlotRange -> All, PlotLabel -> "PlotRange->All"]
  ListPlot[data,  ImageSize -> 300, 
    PlotRange -> {Automatic, {70, All}},  
    PlotLabel -> "PlotRange -> {Automatic, {70, All}}"  ]}, 
 Spacer[20]]