Go with something/someone can be used to indicate your choice:
go with somethig
2. Fig. to choose something (over something else).
I think I'll go with the yellow one. We decided to go with the oak table rather than the walnut one.
My first thought was that with was elided by the speaker. This is fine in casual speech. However, I think you could also simply take go to mean choose, or predict. One way to see this is that we can use go to mean bet:
a. To bet: go $20 on the black horse.
We often say bet something will happen (without actually wagering anything) simply to give a prediction, guess, or opinion.
In other words, go (with) Joshua tells us that the speaker predicts Joshua will be the eventual winner. Late just tells us when he will win. A more formal sentence would be:
I predict Joshua will win late in the fight.