Assuming it’s one of those Green, Libertarian, Socialist campaigns where it’s just someone running for President and no real party presence down ballot.

Let’s say they get enough votes to win. How would a third party candidate govern with a split congress made up of their party opposition? Do they cave into one side? Fight them for four years?

its unlikely a third party could win the presidency without winning a decent amount of the congresses. At that point it would be just like any other time when a president does not have a majority in congress. Assuming both the dems and republicans have a good chunk they could likely work with them and get some stuff done. assuming the republicans stick to their stance of just blocking crap they might be able to slow things down and such.

It's rare for a third party to win more than a single county. Might as well ask what will happen when the seas turn yellow.

The veto is a strong power. And if the Congress is still staffed with do-nothings, then they can rule by executive fiat like the current administration does.

This is a hypothetical

As is the seas turning yellow.

Then why dont you? If youre curious, no harm in asking.