Axes with Ticks in ArrayPlot

Update: Making the two methods in the original answer into functions:

The first method:

ClearAll[aP]
aP = Module[{dt = #, ao = #2, dim = Dimensions[Transpose @ #] + #2, 
          dt2 = ArrayPad[#, {{0, #2[[2]]}, {#2[[1]], 0}}], tcks, inset}, 
        tcks = {Table[{i + .5, i + 1}, {i, ao[[1]], dim[[1]] - 1}], 
            Table[{i + .5, dim[[2]] - i}, {i, ao[[2]], dim[[2]] - 1}]};
        inset = ArrayPlot[dt2, AxesOrigin -> ao, Ticks -> tcks, ##3, 
            Axes -> True, Frame -> False, Mesh -> All, 
            AxesStyle -> Directive[Thick, Red], 
            TicksStyle -> Directive[Thick, Red, FontColor -> Black]];
        ArrayPlot[ConstantArray[0, Reverse@dim], 
          Epilog -> Inset[inset, {0, 0}, {0, 0}, Scaled[1]], ##3, 
          FrameTicks -> ({{#2, #2}, {#, #}} & @@ Range@dim)]] &;

Examples:

SeedRandom[1]
data = RandomReal[1, {7, 9}];

Row[Labeled[Panel@aP[data, #, ImageSize -> 1 -> 25], 
   Row[{"AxesOrigin -> ", #}], Top] & /@ {{2, 2}, {1, 3}, {3, 2}}, Spacer[10]]

The second method:

ClearAll[epilogF]
epilogF[data_, ao_] := Module[{dim = Dimensions[Transpose @ data] + ao}, 
  {Table[{Text[i + 1, # - {0, .25}], Red, Thick, 
       Line[{Offset[{0, 3}, #], #}]} & @ {i + .5, ao[[2]]},
     {i, ao[[1]], dim[[1]] - 1}], 
   Table[{Text[dim[[2]] - i, # - {.25, 0}], Red, Thick, 
       Line[{Offset[{3, 0}, #], #}]} & @ {ao[[1]], i + .5},
     {i, ao[[2]], dim[[2]] - 1}]}];

Examples:

SeedRandom[1]

data = RandomReal[1, {9, 7}];

Row[Labeled[Panel @ ArrayPlot[ArrayPad[data, {{0, #[[2]]}, {#[[1]], 0}}], 
        ImageSize -> 1 -> 25, 
        Mesh -> All, 
        GridLines -> (List /@ #), 
        GridLinesStyle -> Directive[Red, Thick], 
        Method -> {"GridLinesInFront" -> True},
        Frame -> True, 
        FrameTicks -> Automatic, 
        Epilog -> epilogF[data, #]], 
      Row[{"AxesOrigin -> ", #}], Top] & /@ {{2, 2}, {1, 3}, {3, 2}}, 
   Spacer[10]]

Original answer:

SeedRandom[1]
data = RandomReal[1, {8, 8}];

ao = {2, 2};

dims = Dimensions[Transpose @ data] + ao;

ticks = {Table[{i + .5, i + 1}, {i, ao[[1]], dims[[1]]}], 
   Table[{i + .5, dims[[2]] - i}, {i, ao[[2]], dims[[2]]}]};

ap = ArrayPlot[ArrayPad[data, {{0, ao[[2]]}, {ao[[1]], 0}}],
   AxesOrigin -> ao, Axes -> True, Frame -> False, Mesh -> All, 
   AxesStyle -> Directive[Thick, Red], 
   TicksStyle -> Directive[Thick, Red, FontColor -> Black],
   Ticks -> ticks]

You can use Inset[ap, {0, 0}, {0, 0}, Scaled[1]] as Epilog in an empty ArrayPlot to get the desired result:

ArrayPlot[ConstantArray[0, Reverse @ dims], 
 Epilog -> Inset[ap, {0, 0}, {0, 0}, Scaled[1]],
 FrameTicks -> Automatic]

An alternative approach: add ticks and tick labels as Epilog directly:

epilog = Join[
   Table[{Text[i + 1, #], Red, Thick, 
       Line[{Offset[{0, 3}, # + {0, .5}], # + {0, .5}}]} & @
          {i + .5, ao[[2]] - .5}, {i, ao[[1]], dims[[1]] - 1}], 
   Table[{Text[dims[[2]] - i, #], Red, Thick, 
       Line[{Offset[{3, 0}, # + {.25, 0}], # + {.25, 0}}]} &@
          {ao[[1]] - .25, i + .5}, {i, ao[[2]], dims[[2]] - 1}]];

ArrayPlot[ArrayPad[data, {{0, ao[[2]]}, {ao[[1]], 0}}],
  Mesh -> All, 
  GridLines -> (List /@ ao), 
  GridLinesStyle -> Directive[Red, Thick], 
  Method -> {"GridLinesInFront" -> True}, 
  Frame -> True, 
  FrameTicks -> Automatic, 
  Epilog -> epilog]