Yup you should be just fine putting the already established filter on the new tank. I would add just a handful of gups to begin with and if you have test kits check the water parameters. After about a week you could add more fish.
pH is low enough to add some guppies and Prime.
What you are doing is not much different than how I "minicycled" a new 55gal..
Old media, plants, "canaries" (a few guppies). Tank still unbalanced a bit..but worked out w/ no losses to fish.. Plants are another story..
Out of curiosity, what is the pH of the tank the media was removed from, if it is still running. not clear on that.