Not according to this stack exchange answer[0]. It quotes a page from John Hopkins University, although that source page doesn't seem to exist anymore.

> Though New Horizons will also reach 100 AU, it will never pass Voyager 1, because Voyager was boosted by multiple gravity assists that make its speed faster than New Horizons will travel. Voyager 1 is escaping the solar system at 17 kilometers per second. When New Horizons reaches that same distance 32 years from now, propelled by a single planetary swingby, it will be moving about 13 kilometers per second

[0] https://space.stackexchange.com/questions/3520/when-will-new...

Wikipedia disagrees:

> Provided Voyager 1 does not collide with anything and is not retrieved, the New Horizons space probe will never pass it, despite being launched from Earth at a higher speed than either Voyager spacecraft. The Voyager spacecraft benefited from multiple planetary flybys to increase their heliocentric velocities, whereas New Horizons received only a single such boost, from its Jupiter flyby. As of 2018, New Horizons is traveling at about 14 km/s, 3 km/s slower than Voyager 1, and is still slowing down.

Oh, cool