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.
87.3 kilometers per second (sez here) is 196,000 miles per hour.
Light runs about 186,282 miles per SECOND.
The ‘roid is moving about .029120145 percent the speed of light, btw.