How to classify named entities of the same type?
You might be interested in resources built around TimeML, I think there are some corpora and specific parsers specialized in extracting the time details of events. I don't remember any specifics but I tried to google "timeml extract time" and found a few related resources, that might give you at least some inspiration about how people have dealth with similar problems.

If you don't find anything that suits your need, in general the best approach would be to train a custom NER using your own annotated dataset with "start-time", "end-time", "neither" labels.