Why this must be wrong
The Gemara Zevachim 118b derives from Kalev’s statement that he was 40 when he was a spy (Joshua 14:7), 85 when he said those words (v. 10), and 78 when they crossed the Jordan (40+38=78) that the Jews fought for 7 years (85-78). Based on a complicated algebra regarding when the Temple was destroyed (which comes from Seder Olam itself) or a comparison to the fighting, the Gemara concludes that they took 7 years to settle as well. As such, the 28 yers mentioned here must overlap Moses’ leadership - there were only 14 years after Moses died.
Therefore, I suspect that you’re reading a corrupted text or translation.
What the text actually says
You don’t cite where in Seder Olam exactly you got this from, so I’m basing the following based on guesswork. Seder Olam 12 begins with the following statement:
יהושע פרנס את ישראל עשרים ושמנה שנה ויהי אחרי הדברים האלה וימת יהושע בן נון וגו׳ ויקברו אותו וגו׳
Joshua was a provider for the Jews for 28 years. “And it was after these things that Joshua son of Nun died...and they buried him...”
The quote at the end is toward the end of Joshua 24. The important part is the beginning. He wasn’t a judge for 28 years; he was a provider. The word שופט means judge; the word פרנס refers to someone who provides for people’s needs.
Where your text may have come from
The remainder of the chapter, after addressing Elazar son of Aaron’s death, goes down the list of judges. Perhaps the translator assumed that Joshua and Elazar are also judges for that reason.
Further, the following chapter opens up with the following statement about Eli the Priest:
עלי פרנס את ישראל ארבעים שנה
Eli was a provider for the Jews for 40 years.
And this same Eli is identified as having judged the Jews for the same 40 years in I Samuel 4:18:
והוא שפט את ישראל ארבעים שנה
And he judged the Jews for 40 years
So perhaps whoever translates your copy of Seder Olam confused פרנס for שפט for that reason, and since the same word is used by Joshua, that error extended to him.
TL;DR, and questions left hanging
When did Joshua become a judge? Never. When did he become a provider? Fourteen years before entering Israel. What does being a provider mean in a world in which they have food raining from the sky and water flowing from a miraculous well? I’m not sure exactly. But that’s what the Midrash says - feel free to ask, maybe someone can help you out with that piece, because I’m at a loss.
