Wow, I found the general solution right after I posted. It turns out that you need to align it to the top, instead of the right:
radChart.PlotArea.XAxis.Appearance.LabelAppearance.Position.AlignedPosition = Telerik.Charting.Styles.AlignedPositions.Top;
However, I'm still having issues with labels of longer text being mis-aligned compared with labels of shorter text. Is there a way to make sure all the labels are evenly spaced, regardless of text length, but still keep them top aligned?