i am one to recommend replacing or at least rinsing a substrate anytime a system is moved. i am also one that does not like to use substrates at all because of having to keep them clean (though i prefer to keep an oligotrophic (nutrient poor) biotope) you may want to keep a more eutrophic biotope, in which case some substrate can be helpful. just like any other pet, a reef tank needs to be kept clean. one must clean up after the organisms that are kept in the system. just like a birds cage the substrate needs to be cleaned on a regular basis in order to keep it functioning properly. in the case of a reef tank, as a phosphate binder. unfortunately, this idea still seems to be absent in most reef keepers so it is not unusual to come across tanks from other users that have never touched a substrate, what has unfortunately been encouraged in the past. when getting a system that has been maintained by somebody else, you are unsure on where they stand on the methodologies of cleaning up after your pets, it is a good idea to either replace the substrate (if you need one, either for looks, or because there is a certain organism that must need it) or at the very least rinse it clean of all detritus. this way you are sure where you stand on the substrates ability to bind phosphates.
in your case you did not do anything with the substrate, which explains the cyano. how you want to go from here is up to you. if it was me, i would pull everything out and replace the substrate with new dry substrate, or go bare bottom (i say BB because you are unsure on the maintenance in general by the previous owner and being comprised of calcium carbonate, the LR can also be high in phosphates, just not as high as the substrate, i can go into this more if you would like and provide references about how phosphates and calcium carbonate behave together). you do not need to worry about a "cycle". there will be plenty of bacteria in the LR to deal with the nitrogenous compounds. it is the phosphate cycle that you are trying to get ahead of. you can try gravel vacuuming, but that can be tough depending on how coarse the substrate is. if it is a coarse substrate, the it could be pretty easy, if not, then it is going to be a real pain and i would just get new substrate if you feel you need it.
there are not any critters that would help. if there were, then we would not be having this discussion.

: this is one of those myths that is still going around in the reef hobby. that there are critters that help migrate detritus upwards through a substrate instead of downwards. in fact there are two basic kinds of critters that can be in the substrate the predators, and the "detrital eaters". the "detrital eaters" help in migrating the detritus deeper into the substrate (which helps is keeping the substrate viable for as long as possible) by moving the grains around and allowing gravity to pull the detritus deeper into the substrate. the predators on the other hand eat the detrital eaters. not necessarily a good thing. we need the detritus to migrate downward. eating all of the detrital eaters in the top few inches of a substrate can cause the substrate to become clogged with detritus leading to dead areas, which then lead to anoxic areas, which then leads to the dreaded hydrogen sulfide. the LR will provide the critters needed to keep the substrate functioning pretty well. the thing to remember is that all organisms poo. i have yet to see an organism with its mouth deep in the substrate and its butt in the water column. until this organisms if found nutrients are always going to build up in a substrate. in other words. one must clean up after all of the organisms that one keeps. whether it is cattle, and the ranger rotates the grazing areas to keep the poo away from the cattle, or a bird in a cage, where the birder replaces the newspaper on the bottom, or a reef keeper keeping bristleworms and nematodes in a substrate and siphon detritus or replace the substrate on a regular basis. not cleaning the substrate on a regular basis is a lot like not taking out the trash in the house ever and letting the roaches, maggots, rats, and mice take care of it, or never flushing the toilet and hoping that putting a potted plant in the toilet is going to make the wastes go away. we flush and take out the trash, why should our reef tanks be any different? we all must clean up after our pets.
have you thought about what critters you are wanting to keep?
G~