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> a slight push in the direction of the sun

That's not how orbital dynamics work; a slight push in any direction won't change the orbit enough for that thing to not be a problem anymore.

To make something fall into the sun, one needs quite a bit of delta-vee, which means energy. However, if the thing isn't in any hurry (and I imagine waste isn't), one can simply attach a solar sail with a tiny computer for steering it, and let it brake for the next few thousand years.

(Kerbal Space Program should be part of school curricula)

It's actually cheaper (in delta-V terms) to leave the Solar System than it is to hit the Sun [1].

[1]: https://www.reddit.com/r/askscience/comments/5lt50r/why_does...

Pushing something into the sun requires removing most of it's orbital velocity not just a simple push. A small push in an orbit will create something on a very similar orbit to you that absent a nearby body to induce a lot of drag will intersect with your orbit in 2 places (not at the same time you're there fortunately in pretty much every case).