I don’t think atmospheric drag helps much, since as mentioned, you’ll go through it over and over. Even if you use it (somehow) when it is most expanded, it’s not like there’s a sharp cutoff beyond which there’s vacuum.
If you can wait 18 years and change, you could perhaps use Apophis’s gravity to adjust your orbit.
Using the Moon and Lagrange points, maybe one could do it. Start reading about The Interplanetary Transport Network. Maybe if you hit one Lagrange point, with an amazingly accurate (to say the least) aim so that the object then heads on to another one, etc. Note that for Earth orbit, it would still be a lot more efficient to just get to space and then do an orbital correction burn. But if you dream big of interplantery travel, then this could do it.
Those are all inherently unstable orbits. The instability is a feature, not a bug, for interplanetary transport, because it means that small initial changes in the orbit (i.e., needing small amounts of fuel) can lead to large changes in orbit later on (like reaching an entirely different planet).