Above is correct, the water contains not much beneficial bacteria from what I understand. Transferring an aged sponge filter should be enough, and I have use this method many times, with a few caveats
1) the bacterial population may take some time to adjust to the level of fish in the new tank- if you put one small sponge in a large tank with heavy bioload, it may not be enough and you may get a "mini cycle" with visible levels of ammonia, nitrite, nitrate until the BB population has time to grow.
2) the sponge will need to have been established for a while, i'd recommend over a month, if you just put it in the other tank for a few days and then transfer it over, there won't be much bacteria on it.
3) you need to keep the bacteria alive. It will need some ammonia, nitrite/ate in the new tank already to keep it's population high. For this reason, your idea of transferring 25% of the water from the old tank is not a bad one, although it may not be necessary if you do have enoughh bioload in the new tank to quickly start feeding the bacteria.
With all that said, I've done this many times (haven't cycled a tank "the old way" in decades) , and it usually works without much worry. If yo do get a "mini cycle", you can just do a small partial water change to keep levels low enough not to kill your fish. Best of luck!