This should at a minimum be possible in Catalog feature class properties with arrow buttons, but in the year 2015, this really ought to be drag-and-drop functional. Seems long overdue.
My definite vote - extremely useful tool particularly when other editors may have deleted specific fields you actually required. The manual process to fit them in again would involve re-creation and re-loading, yet with this you could simply add fields and then re-order.
If I have to re-order fields, I use X-Ray for ArcCatalog.
Chris Rumig | GIS Analyst – Information Technology Services | Region of Waterloo
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I use the Table Restructure Tool in the XTools Pro extension to accomplish table reordering and other schema updates. Not to excuse Esri from including this functionality in the base ArcGIS product but they do create an opportunity for third party developers to make a living.