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]]

[![enter image description here][1]][1]

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}}"  ]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/YoVBL.png
  [2]: https://i.sstatic.net/bBSMe.png