Yes, if there is a filter on your 10-gallon shrimp tank that contains biological filtration media (bioballs, sponge or foam, ceramic rings, etc.) for the beneficial nitrifying bacteria to grow on, then you should be all set when you are ready to cycle the tank. The one small piece of live rock in the aquarium is more than adequate to seed the tank with beneficial Nitrosomonas and Nitrobacter nitrifying bacteria, and adding a piece of raw, uncooked cocktail shrimp to decompose will provide a source of ammonia to feed the nitrifying bacteria so that a large population of them grows and comes to inhabit the biological filtration media.
As you know, you can then monitor the cycling process by following the rise in the ammonia level, followed by a subsequent rise in the nitrate level as the ammonia peaks and begins to decline. When both the ammonia and the nitrite levels have dropped to zero, and nitrate begins accumulating in the aquarium, then you will know that it has cycled and the biological filtration is established. It will be able to support the red feeder shrimp once that happens.
So all you need to start the cycling process is for the tank to have biological filtration media to act as a substrate for the beneficial bacteria, a bit of live rock to seed the tank with the nitrifying bacteria, which you already have, and decaying cocktail shrimp to feed the beneficial bacteria with ammonia. Other than that, it’s just a matter of time for the nitrogen cycle to be completed and establish the biological filtration.
However, the red volcano shrimp will appreciate having several fist-size pieces of live rock in their aquarium to provide them with shelter, something to crawl around on, and act as a source of food as microalgae grows on the rock. The red feeder shrimp (Halocaridina rubra) won’t feel comfortable or breed well in a bare aquarium that offers them no shelter or structure to crawl on.
For best results, I would order some Shrimpgro and Ulva sea lettuce to feed the volcano shrimp in the newly established aquarium. The new tank will not have time for a significant amount of macroalgae or fungus to grow, or for a film of bacteria (biofilm) to build up on the surfaces in the aquarium, which are what the volcano shrimp feed upon. The Shrimpgro and sea lettuce will suffice to feed the volcano shrimp as the aquarium matures and stabilizes.
Best of luck cycling your 10-gallon aquarium and maintaining a thriving colony of red volcano shrimp!