This is not the most elegant method. But if you already run Arc Hydro, it wouldn't take much to do this for one or two reaches. If you want to perform this for several points, repeat it over and over, or code it, you will want to search further.
If you have a flow direction grid, you can use a Arc Hydro toolbar tool that will put a graphical line on the flow path downstream from the point you click. You can then convert that graphic to a feature using the Drawing toolbar menu command. After that you can use other ArcMap tools to measure or trim that polyline to a certain length.