Ah, you have a 10 G already cycled ? Then all you have to do is seed the new filter on the new tank with the rinsings from the 10G filter. When you clean it, in tank water, pour the dirty tank water directly into the new filter. You can do that a couple of times, don't overclean the old filter, but if you do it every couple of weeks, dose the new filter twice.
Meantime, use ammonia to feed the new filter, as without fish the BB do die off fairly soon. You might have a cycled filter in a week, it really depends how much BB washes off when you clean the other filter. Shouldn't take more than two dosings though, and then you can likely begin adding fish, SLOWLY.. test to make sure the cycle is complete, zero ammonia, zero nitrite, moderate nitrates under 50ppm, and then you can add one or two fish, and not more than 3 weekly 'til you have the stock you want.