You tank will NOT cycle. There should be enough bacteria in your substrate, decor, etc. to help with the bioload. Just keep up with the water changes very frequent. I would add a cheap powerhead with a venturi system to keep the tank well oxygenated.
I'm unfamiliar with your filter, but if you have a sponge media that was in your filter, remove it and leave it in your tank. The bacteria on the media will remain. Then when you new filter arrives, use the sponge in there if is possible. Otherwise, after several weeks, you can remove the sponge media in the tank.