The filter will have to be on as it is actually the filter you are cycling and not the tank or water but you should look for other filter media such as SeaChem Matrix to replace the carbon. Carbon is primarily a chemical filter more of use when removing medications (which hopefully, you'll never need). It has to be changed every couple weeks and when you change it, you end up throwing a large portion of your beneficial bacteria away since they primarily colonize the filter media.
As mentioned, look for ammonia in the cleaning section of your local grocery store or home improvement store. I'm assuming you are in the US since your filter appears to be a HOB and they aren't very common elsewhere. If so, Walmart usually has ammonia. Just make sure it doesn't have detergents, perfumes or surfactants in it. Basically, if you shake it and it foams, it won't work.