Will the current sponge in my filter instantly cycle a new tank/filter?

Am I able to use my very established sponge that’s in the current filter, in a brand new filter, to instantly cycle the tank? (it came with a new filter) or can i use the old filter to instantly cycle?
Noticed a small leak in my tank so immediately bought a new one, it’s gradually getting worse so it’s pretty urgent & I would like to switch everything over asap. Do I need to add the original water into the new tank?
Any help on the proper steps would be greatly appreciated. I did a lot of internet research but figured I’d ask real people with experience, in hopes of the best possible outcome for my fish. Thanks in advance!

Tank size - 36 gallons
Capacity -
Three adult black skirt tetras
Three tiny baby black skirt fry
One bristle nose pleco