I am just curious to know if there is any estimation, no matter how rough it could be, about when another human artifact can go farther than Voyager 1 with respect to the Earth; years, decades, any planned mission?
Voyager 1 is currently travelling at 17 km/s, or about 3.5 AU per year. Anything that would overtake it would have to be moving faster than this away from the Sun.
There have been proposals to fly probes faster than this - the idea being that if your primary mission is to study the interstellar medium or the heliosphere, you want to be able to get there as quickly as possible.
There are two future missions (one planned, one proposed) that might do this.
Firstly, there are a pair of Chinese "Interstellar Express" probes, with a planned launch date in 2024; the WP summary doesn't give an exact speed, but a back of the envelope calculation based on the dates and distances given there suggests that after the Jupiter encounter, IHP-1 would be moving about 4 AU away from the sun per year, and after the Neptune flyby (exciting!) IHP-2 would be moving about 4.8 AU away from the sun per year.
(The actual velocity would be higher; it would be moving at a tangent rather than directly out of the system. But distance to the sun is what we want.)
Voyager 1 has a substantial head start, of course, so it would take these probes 100-150 years to overtake it.
Secondly, a recently proposed NASA mission, "Interstellar" (detailed proposal), is targeting a very fast trajectory (6-7 au/year) to reach 100 AU after fifteen years and 344 AU after 50. (I think these are solar distances not distance travelled on the tangent, but it's not quite clear.) This would probably overtake Voyager 1 in 20-30 years from launch, which would be in the 2030s - though it is as yet only a proposal.
Voyager 1 is now 154.95 AU away from Earth. The distance increases by 3.6 AU by each year now.
For simplification, I assume constant speed for the whole distance. If we want to start another space probe to overtake Voyager 1 after decades or centuries, the speed should be substantially higher.
A probe with a speed of 4 AU per year will overtake in 433.3 years. With a speed of 5 AU/year, it will take 153.5 years. If we may double the speed to 7.2 AU/year, it will take 86.7 years.
But doubling the speed would require a very heavy and large rocket. Two times the speed means four times the energy, so the rocket needs to be more than four times as large and heavy.