If your lines are split at the fittings, and you have and Advanced license, you can use the Feature Vertices to Points tool. You can run the tool twice, once with the tool option "Point Type" as "Start" and once with "End". You will then have points that coincide or are very close to your existing points. Then use a spatial join to pull the attributes from the existing points to your newly created points. Then you can use an attribute join to join those back to your line (the Feature Vertices to Points tool retains the original FID so you can join it back).
Also, if you don't have an Advanced license, XTools has a tool with that functionality, and there is a free 14 day trial.