Try this (if you have an arcinfo license):
1. Run Feature Vertices To Points tool (Data Management toolbox - Features toolset) uisng the BOTH_ENDS option to get endpoints of all lines.
2. Run Unsplit Lines tool (in the same toolset) to merge lines at pseudo nodes (where only two lines connect).
3. Run Split Line At Points (in the same toolset) to break the merged lines at the endpoints; the resulting lines should be the same as the original lines, hopefully in continuous order.
4. Run Spatial Join (Analysis toolbox - Overlay toolset) with SHARE_A_LINE_SEGMENT_WITH option to join the split lines with the original lines to get the attributes through Field Map.
Does that work?