Lincoln died 150 years ago. I think this is evidence that language changes with the times. Read a little Shakespeare, and you'll find a lot of things that aren't said quite as they would be today.
Having checked the larger context I think you can safely take this as a slightly out-of-date equivalent of "rise to the occasion".
Is "rise with the occasion" emphasizing "to react at the same rate of the progress" more than using "to"?
Read a little farther in the letter to Congress (not a speech) and you'll see that he's arguing that progress should be made, not that Congress or the country should try to keep up with progress that is already being made. From the letter:
Is it doubted, then, that the plan I propose, if adopted, would shorten the war, and thus lessen its expenditure of money and of blood? ... We can succeed only by concert. It is not "can any of us imagine better?" but, "can we all do better?" The dogmas of the quiet past, are inadequate to the stormy present. The occasion is piled high with difficulty, and we must rise -- with the occasion. As our case is new, so we must think anew, and act anew. We must disenthrall ourselves, and then we shall save our country.