Voyager 1 received gravitational assists increasing its velocity from both Jupiter and Saturn.
Opportunities for such a Jupiter-Saturn trajectory occur about every 20 years, so we don't have to wait too long to launch, at least; in order to outperform Voyager we merely need to launch a probe of about the same total mass as Voyager, at the same initial velocity, with the same gravity-assist parameters, but including a propulsion system in place of the 105kg of scientific instruments carried by the earlier probe. An ion engine would give the most ∆v for the mass; something like an NSTAR thruster of 25kg plus 80kg of Xenon propellant, on a total spacecraft mass of 773 kg, firing for about a year, would yield an additional 3km/sec above Voyager's speed. (Power would be a bit of a problem; Voyager produced only ~430W from its radiothermal generators, while NSTAR wants 2100W for maximum efficiency, but we can handwave this away by assuming a smaller thruster operating for a longer period of time.)
Improvements in computer and other technologies over the last 40 years should make it possible to shave off additional mass from the probe, of course. Those mass savings could pay for additional propellant on the probe.
The Titan IIIE booster that launched Voyager is no longer in service; the Atlas V 551 configuration that launched New Horizons slightly outperforms the Titan in mass-to-LEO, so I assume it could serve this role.
Voyager 1 does have a substantial head start, however, so it could take something like 200 years for this speedy little probe to outdistance it.
A larger booster -- Delta IV Heavy, Falcon Heavy, SLS, or SpaceX's future rockets -- could of course lift a very powerful upper stage that could give the probe another substantial boost, perhaps cutting the outdistancing time in half or further.
So, with no major technological advances, I estimate it should be no more than 20 years from budget approval to launch, and 100-200 years from launch to outdistancing depending on the budget.