Convert Derivative[1][f] to f'

You can use TemplateBox to control how derivatives are copy/pasted:

MakeBoxes[Derivative[n_Integer?(Between[{1,4}])][f_], StandardForm] := With[
    {p=StringRepeat["\[Prime]",n], q=StringRepeat["'",n]},
    TemplateBox[
        {MakeBoxes[f]},
        "Derivative1",
        DisplayFunction->(SuperscriptBox[#1, p, MultilineFunction->None]&),
        InterpretationFunction->(RowBox[{#1, q}]&)
    ]
]

MakeBoxes[Derivative[n__Integer][f_], StandardForm] := TemplateBox[
    {MakeBoxes[f], RowBox[BoxForm`MakeInfixForm[{n}, ",", StandardForm]]},
    "Derivative2",
    DisplayFunction->(SuperscriptBox[#1, RowBox[{"(",#2,")"}],MultilineFunction->None]&),
    InterpretationFunction->(RowBox[{RowBox[{"Derivative", "[", #2,"]"}],"[",#1,"]"}]&)
]

One approach is to override Derivative box formatting and use Copy As -> Plain Text

MakeBoxes[Derivative[n_Integer][f_], form_] := RowBox[{ToBoxes@f, StringRepeat["'", n]}]