If Smash Run isn't online, I indeed have a lot less use for it. I will still pour many hours into it, as it is the best grinding mode in the game, and it's a very nifty grinding mode. But I was hoping it would be a lot more than that for me.
And the annoying part is that your rewards are BETTER if you play with human players. So that's really stupid if it isn't online.
However, even if it is not online on release, it could be patched in. In Tekken 6, there was a special single-player mode called "Scenario Campaign", basically the Subspace Emissary of that game. However, later on, Bandai Namco patched in an online 2 player co-op version of the mode, with 10 redesigned and very lengthy stages, and it was really good for getting fight money and customisation pieces in that game.
As it is, everyone has their own version of the map in Smash Run. So I really don't see syncing as an issue. All that needs to be synced are the alerts, map icons, and stats in the Smash Run map. This is exactly how local play works, so I don't see how it can't work online. The actual syncing of what fighters are doing only needs to be done in the final battle.