Assuming Kratos can perceive Stands, he certainly still won't solo the verse. He has exactly zero answer to Stands like Gold Experience Requiem or Stairway to Heaven (infinitely better name than Made in Heaven).
He'd easily get through part 1 and even 2, since though he can't kill Kars he could replicate BFRing him pretty easily, being physically far above any Jojo character I know of and perfectly capable of chucking Kars into space.
Part 3 is where things get a little more interesting. Kratos maintains this physical superiority advantage over even the most physically dominant Stands in this time, like Star Platinum and The World. Though they are apparently faster due to them being FTL, Kratos could take Dio's MUDAMUDAMUDAMUDA barrage with absolutely no injury. And though the Stands are faster than Kratos, their wielders aren't. Kratos could kill Dio and Jotaro with casual ease, and he's actually resistant to time stops so ZA WARUDO won't work very well either.
Even Stands Kratos couldn't conventually fight, like The Lovers, would be rendered moot by Kratos' willingness and ability to simply kill their wielder, since the format of the thread is in 1v1 scenarios.
Alessi's Stand is broken enough to actually affect Kratos, but I honestly think that, like with Jotaro, Kratos would be capable of beating the shit out of Alessi even if turned into a child.
As stated, Kratos would defeat Dio in all likelihood IMO, so this leaves Vanilla Ice as the only one who could potentially beat him. His Stand is extremely powerful and could easily kill Kratos. If Vanilla Ice starts the fight inside his Stand it would also be hard for Kratos to injure him, since touching the Stand would cost him whatever he used to touch it, it being a ball that negates space and essentially ignores conventional durability entirely. Its only weakness is that Vanilla Ice can't actually see when inside the Stand, so he leaves himself vulnerable if he wants to get a look at his surroundings. That said, there's no reason to assume that Ice would start out inside the Stand, so this point is moot, and Kratos would sort of hard counter both him and Dio with the Head of Helios, which emanates sunlight that would instantly kill them, since they're vampires. This would also work on the Pillar Men of Part 2.
I don't know enough about the other Parts to really say how Kratos does in general, but I know he gets stopped hard by Giorno Giovanna's Gold Experience Requiem. Gold Experience Requiem first of all reacts to all threats, even onces like universal time stops, and sets all attacks back to 0. This makes Giorno impossible for Kratos to injure. It can also trap someone in a state of repeating death for all eternity. It's one of the most broken Stands in the canon, and Kratos has no answer for it. He loses.