What major orbital problem of the late 1800s is solved by general relativity?

That would be the precession of Mercury's perihelion.

In simpler terms this would mean that the Mercury's orbit, perihelion, did not behave as it should according to the Newton's laws.  That is its rotation of the orbit, the precession, was strange as it was at its closest to the sun at different places as it slowly moves around the sun.