いけちょ(@ikeda_shogouki)さんのコードを少し変更してみました。

gistfile1.cs

        <Grid HorizontalAlignment="Right"
  		  VerticalAlignment="Center"
			  Margin="0,0,20,0"
			  RenderTransformOrigin="0.5,0.5"
			  Width="{Binding ActualWidth, ElementName=innnerTextBlock, Mode=OneWay}"
			  Height="{Binding ActualWidth, ElementName=innnerTextBlock, Mode=OneWay}">

            <Grid Background="Blue" HorizontalAlignment="Center" VerticalAlignment="Bottom">
                <TextBlock Name="innnerTextBlock"
					       Foreground="Black"
					       TextAlignment="Center"
					       VerticalAlignment="Bottom"
					       FontSize="24"
					       Text="ここには可変長のテキストが入る" />
            </Grid>
            
            <Grid.RenderTransform>
                <CompositeTransform Rotation="270" />
            </Grid.RenderTransform>
        </Grid>