That's usually what most people do when upgrading their tanks, just run the new tank on its own for a few weeks (usually 4-6 weeks to let it cycle), best if you can transfer over some bio-media from the older tank to "seed" the new tank with beneficial bacteria, helps speed up the cycle.
In your case, with just an air pump, there may not be as much media surface area for the bacteria to grow on initially (just on the tank's glass surfaces and soil), so the cycle may take abit longer.
The initial ammonia and nutrients released from ADA aquasoil is designed to boost plant growth during the startup period, if you have no plants in the tank, then just do daily large water changes (ie. 80%-90%) for 1-2 weeks to flush out the ammonia, the ammonia levels will gradually lower to more manageable levels for the bacteria to handle while the cycle is progressing.
Btw, it'll be abit of a waste to just throw out all the nutrient rich water, so if you have potted house plants, use the tank water to fertilize them... save $$$ on ferts and save water too.
Alternatively, you could just add lots of fast growing plants in the new tank. The plants can utilize all the rich nutrients from the soil and grow (no need to do so much tedious water changes) and in this way you can cultivate more plants for use in the future too. Later you can remove or re-position those plants when you transfer the older tanks plants and hardscape over to the new tank.
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