Learn Arabic (Modern Standard): في الكوب حليب وسكر فقط.

Arabic doesn't have a direct equivalent for 'there is' or 'there are' in the present tense. Instead, it places a location phrase like في الكوب (in the cup) at the beginning of the sentence. Putting the location first before indefinite nouns like حليب (milk) automatically gives the meaning 'There is [noun] in [location].'

Since the current lesson is about the number 1, why don't we use واحد to say 'one milk' or 'one sugar'?

The words حليب (milk) and سكر (sugar) are uncountable nouns. Just like in standard English, you wouldn't use 'one' directly with them unless you are talking about specific units, like one cup of milk (كوب واحد من الحليب). Because we are just talking about the substances in general, no numbers are used here.

Why is the word for 'only' (فقط) at the very end of the sentence?

In English, we usually put 'only' before the things it describes ('only milk and sugar'). In Arabic, the word فقط (only/just) typically comes after the word or phrase it applies to. So, the sentence literally flows as 'in the cup [is] milk and sugar only.'

Why does the cup (الكوب) have ال (al-) but milk (حليب) and sugar (سكر) do not?

The cup is a specific object ('the cup'), so it takes the definite article ال (al-). Milk and sugar are general, unspecified amounts here, so they remain indefinite (without ال). Using an indefinite noun after a location phrase is exactly how you create the 'there is...' meaning in Arabic!