You would think Eric Berger (who's a pretty seasoned space writer) would have played Kerbal Space Program. That game took my understanding of orbital dynamics to a whole other level. I was immediately bothered by that paragraph as well.
I'm not a KSP pro, but I have tried and tried to fly into the sun and have yet to succeed. Even if I do my best to lose as much of the planet's orbital velocity as I can until I'm out of fuel, and I begin to fall towards the sun.... I still always miss and then just go into an elongated elliptical orbit. It's really hard.
You'll want a bi-elliptic transfer orbit. And probably a larger rocket.
I would suspect that a slingshot at Jool would do it but I've never tried.