I'm just nipping down the road to get a paper.

It suggests that the activity is minor or simple, will be done straight away, doesn't take a great deal of time, that kind of thing.

Does the sentence refer to a present or a future? I think it refers to the near future - Am I right?

Probably, but we'd need more context to be sure.

I don't know the context. I've found the sentence in the dictionary http://dictionary.cambridge.org/dictionary/learner-english/nip

Then we cannot be certain of the time of the action.

AnonymousWhat does "just" mean in the sentence below?

It's there to convey the idea that the action is a trivial one.

CJ

AnonymousDoes the sentence refer to a present or a future? I think it refers to the near future

Situation 1: An announcement at the breakfast table: I'm just nipping ... (This indicates an action about to happen in the near future, similar to I'm just going to nip ...)
Situation 2: On the way to get the paper, you are stopped on the street