Learn Arabic (Modern Standard): غسل الولد كوبا واحدا.

To ask 'how many', you use كم (how many) followed by a singular noun with an 'an' sound at the end. You would ask كم كوبا غسل الولد؟ (How many cups did the boy wash?). Notice how كوبا (a cup) takes the exact same spelling in the question as it does as the object of this sentence!

Why do both كوبا and واحدا have an extra alif (ا) at the end?

That extra alif is added because these words are the object of the verb (the thing being washed) and they are indefinite (they do not start with 'al-'). In Arabic, when an indefinite word is the object of a sentence, it takes an 'an' sound at the end, which is written with an extra alif.

Why does the verb غسل (washed) come before الولد (the boy)?

In Arabic, sentences usually start with the verb. You literally say 'Washed the boy one cup.' You could also put the boy first, but starting with the verb is the standard and most common word order for actions.

Doesn't كوبا already mean 'a cup'? Why do we need to add واحدا (one)?

You are right that كوبا alone means 'a cup'. Adding the adjective واحدا is used for emphasis. It strongly stresses that he washed exactly one single cup, rather than two or three.