I'd recommend buying new biorings (or whatever biological media you use) and adding it to the filter of the established tank for a few weeks. Let the nitrifying bacteria colonize the new media. Then transfer seeded media to the new tank. It should theoretically be cycled.
Just take into account that the amount of bacteria in any given system is limited by that systems bioload. So if you have let's say twenty neon tetras, all the surfaces in your tank, including the filter media, will have enough bacteria to support twenty neon tetras. The point I'm trying to make, is to not overload a tank when cycling in this method because you are limited by the bioload of the original tank.
This said, you shouldn't have any issues with doing this for a small shrimp tank as their bioload is small. You can use sponges, biorings, anything bacteria will grow on. I prefer using biomedia as it feels cleaner. I always have spare biomedia in all my filters.
Another thing to note, plants, especially floating plants, will use up ammonium. If your tank is heavily planted, you can add the shrimp and probably won't even need the to cycle the tank. In any case, I do recommend both, better be safe than sorry.
Edit: Using "old water" from another aquarium does not cycle the aquarium because the nitrifying bacteria live on surfaces not in the water column.