Perhaps something like this? You may have to do a little tweaking of the parameters:

With[{imageSize = 500},
 Pane[
  Column[{
    Plot[x, {x, 1, 3}, ImageSize -> imageSize]
     Row[{TextCell[
        "I've created a plot and a caption for it. The caption is a \n    few lines long. I want to size the caption so that it is the same \n    width of the plot and then place it under the plot. I would then \n    export the two combined as a PDF.", "Text"]}, 
      ImageSize -> imageSize + 30]
    }
   ],
  ImageSize -> imageSize + 50]
 ]

And you can change the "Text" in the TextCell CellStyle to whatever other CellStyle you might wish. Or you can add options to the TextCell to control aspects such as the FontSize as in:

With[{imageSize = 500}, 
 Pane[Column[{Plot[x, {x, 1, 3}, 
      ImageSize -> 
       imageSize] Row[{TextCell[
        "I've created a plot and a caption for it. The caption is a \
few lines long. I want to size the caption so that it is the same \
width of the plot and then place it under the plot. I would then \
export the two combined as a PDF.", "Text", FontSize -> 10]}, 
      ImageSize -> imageSize + 30]}], ImageSize -> imageSize + 50]]