This. We have the technology to go this fast... but no pressing need. Should a pressing need arise I do not doubt we will begin immediately. SpaceX could be turning around Falcon Heavy after Falcon Heavy as we loft fuel tanks to a “probe” we assembled in orbit out of an existing space worthy hardware “core” capable of autonomous operation such as a Soyuz or Dragon capsule, any instrumentation could be just built as stand alone modules and bolted to a separate set of communication hardware. We could conceivably build a probe and chase this thing down to contact in under 5 years.

If it took 6 months to build (!!), could we go fast enough to catch up to it in a reasonable time?

Just thinking about this scenario is enough to give me goosebumps.

Imagine the alien probe about to leave the outer perimeter of our solar system while our makeshift-to-the-aliens probe is trying to catch up to it while detonating one nuke after another.

You'd think that would catch their attention.

"See look at that propulsion system! I told you they were f'ing crazy. Fire the engines again for another burn, quick. Let's get some distance before they figure out that you can create a quantum singularity and the whole solar system goes up." - the aliens

NERVA? nah. I'd probably go as far as getting the dust of the Project Orion and Medusa spacecraft. That's how you get there fast.