I believe there is a special cartridge out there, similar to the DS Download Station, that was designed to connect to Nintendo's servers and download Nintendo Zone data, then broadcast it out in the local area (would be running in a special DS/DSi in a store somewhere, for example).
I think this cart was only ever used for the DSi version though, and was basically just the DS Download Station but was able to be remotely updated instead of needing a new cartridge every time they wanted to put new demos out. The cartridge is now useless though, as the servers have been shut down for a while.
As for the 3DS app, it was a built in system app, or at least it was in EUR region for me. It would pop up on the home menu whenever you entered a "Nintendo Zone". I only ever used it a few times, but it basically just had some advertisements/videos for upcoming games and Nintendo products. I don't recall any demos being playable, but maybe I just forgot.
Even if you get the app however, it will be useless without the Nintendo Zone servers, which have long since been shut down. I don't think it would let you any further than the title screen without being in a zone.