This paper explores the possibilities for such a mission. It’s not that far fetched technically. But realistically, I don’t think it would happen, even if the object looked more interesting than it currently does.
Wasn't there a practical design using atomic bombs to propel a spacecraft? Alien artifact would certainly justify the budget and risk. Though if a country determined it was alien, they'd probably not share it until their craft was well on the way.
"Practical" in the sense of yes, the physics work, for the ones we know of. And assuming there was reason to quiet the Green crowd from freaking out about us launching nuclear bombs into orbit.
Knowing it's trajectory and throwing something at it fast enough are two very different things. There's no planets between here and there that we could sling-shot off of. Even if there were, we don't have anything to send out to it.
You can slingshot off of anything in any direction if you can compute your exit vector correctly. A couple of loops between the Earth and Moon and you could pick up enough speed to get anywhere you want.
The trouble is computing these trajectories is pretty tricky and if you get even one manouver wrong you crater or burn up in the atmosphere.
Fastest object we've ever made was Juno, which went 25miles/sec (5 years to Jupiter)... this is traveling at ~30 miles/sec, and it's already getting pretty far from us: https://en.wikipedia.org/wiki/%CA%BBOumuamua#/media/File:Oum...