Is it sufficient to give a portion of zakat to someone who later turned out not to be in need of it, as was initially expected, given that he received the amount after his treatment was provided for free?

It is not permissible to pay Zakat while doubting the recipient's eligibility. Rather, there must be certainty or strong conviction that they are among its rightful beneficiaries.

If the Zakat payer gives Zakat while doubting, and did not investigate, or investigated but it was not clear that the recipient was eligible, then the payment is invalid.

As long as the man underwent the operation and did not need Zakat money for it, Zakat should not be given to him merely based on the assumption that he needs medicine or similar items. If you do not know or are not strongly convinced that the man is among the Zakat beneficiaries, then your obligation has not been discharged, and you are required to pay Zakat now.

If your neighbor transgressed by giving the money to the patient after you informed him that the patient was not among the rightful beneficiaries, and he kept the money then gave it to him without your permission, then he is liable, and you have the right to demand the money from him.