Learn Russian: У них нет восьмидесяти рублей на этот платёж. - They do not have eighty rubles for this payment.

Because the word нет requires the genitive case. The nominative is восемьдесят. For numbers from 50 to 80, both halves of the word decline. So, восемь changes to its genitive form восьми, and десят changes to десяти, making восьмидесяти.

Is рублей in the genitive case because of нет or because of the number 80?

Technically, because of both! In the nominative case, 80 forces the noun into the genitive plural (восемьдесят рублей). However, when a number is in an oblique case (like the genitive required by нет here), the noun agrees with the case of the number. So рублей is in the genitive plural to match the genitive восьмидесяти.

Why do we use на этот платёж instead of для этого платежа for 'for this payment'?

When you talk about what money is going to be spent on, Russian uses на followed by the accusative case. While для means 'for' in the sense of the recipient or beneficiary, на is used to indicate the specific target or purpose of funds.

What is the difference between платёж and оплата?

Платёж is a specific, countable 'payment' or 'transaction' (like a bill or a loan installment). Оплата usually refers to the general process of paying (like 'payment upon delivery'). Since this sentence talks about a specific charge, платёж is the right fit.