meaning of 'slip up on'

The specific phrase "slipped up on" means "made a mistake about" or "made a mistake in regard to." The phrase following "on" indicates exactly what the mistake concerned. In this use, "on" does not imply "during" or "while" as it can in other constructions.

He slipped up on just one detail.

One detail was incorrect, although everything else was right. Often this would be followed by further text giving just what detail was wrong, and how.

Someone had slipped up on the order.

Some mistake was made by someone about the order. Again, this would often be followed by details.

• He slipped up on timing - he should have entered the order before he started to work on it.

• She slipped up on manners -- one does not say "you're welcome" before one is thanked.

While X on Y is used to express X being on top of Y if Y is a physically flat surface, or X is "stuck" to Y somehow, it can also be used a bit more abstractly to express that X is currently or was talking about or dealing with Y as a topic or thing to work on.

The class is currently working on chapter 4 in the text.

It's definitely used to communicate "while X, Y happened":

We were on the 3rd round of the fight when he called = While we were [watching] the 3rd round of the fight, he called

So

He slipped up on an order = While he [was processing] an order, he slipped up.