SpaceX Falcon Super-Heavies and gravity slingshots around Jupiter and the Sun could be used to catch up with the interstellar object. A space mission to chase down the Oumuamua object would be worth the $2 to 4 billion or so cost. Oumuamua is worth looking at for scientific value even it is only an anomalous interstellar object. Harvard researchers have calculated that unusual speedup and movement of Oumuamua indicate that it could be a solar sail.
The fast solar slingshot mission would be a very fast flyby. It would be better to brake.
The Russian company Fakel has sold Stationary Plasma Thrusters (SPTs) for a couple of decades. The SPT-140D masses 8.5 kg, 4,500 W power, a specific impulse of 1770 seconds, a thrust of 0.29 newtons, and a lifetime of 4500 hours. The jet-efficiency is ~56 %.
The SPT-230 has been tested up to 3,200 seconds Isp and 15 kW jet-power and has an efficiency of about ~75%. They are aiming for a 9,000 hour life-span, an Isp of 2,700 seconds, a mass of 25 kg. With a power of 15 kW and efficiency of 0.75 that’s a thrust of ~0.85 N. Constant operation would require 1,040 kg of propellant per engine.
A small probe with about the same mass as the propellant would have a delta-vee of 18,353 m/s total. This would allow braking to match the speed of the interstellar object.
Kilopower fission systems would be great for powering deep space probe missions. Year-long high ISP propulsion from plasma thrusters and kilopowered probes would be revolutionary for deep space.