It seems like this would have been worth a mission to get a closer look, and perhaps plant a communications package on it, something like hanging a bell on a cat. They did it with a comet, why not this?
If there had been enough lead time, possibly, but it would take some serious foolin’ around, because (sez here) it was moving at 87.3 kilometers per second.
The Voyager I (sez here) is traveling faster than Voyager II, but at a speed of about 17 kilometers per second
The current record for fastest (Earth-built) spacecraft (sez here) is the Parker Solar Probe, 532,000 km/HR (147.77 km per sec) at perihelion, which is far in excess of its average velocity, and accomplished via gravity assists and trajectory alterations.
In short, if we could use a rocket to catch Oumuamua to plant a probe on it, we could just launch probes in all directions at any old time, and skip trying to hitch a ride.
Cometary missions require something like 15 years lead time before launch. More years to intercept. We had something like six months awareness of this object from noticing it and its leaving the solar system. No time.