There's some history when it comes to ETH/ETC replay protection.

How was this problem resolved for Ethereum?

Initially when ETH hard-forked to invalidate the DAO contract, both ETC/ETH networks did not implement replay protection for transactions. Instead, a split contract was written instead which relies on a split oracle built in Ethereum to identify if the contract was called on the hard-fork (i.e. ETH chain).

The contract logic is quite simple as you can see for yourself, it basically splits the original tokens and sends them into two different addresses depending on which chain the contract is called.

Nowadays, the ETH (hard-forked) chain has effectively implemented replay protection. So transactions executed on the ETC network will not be valid on the ETH network and vice-versa.

Does Ethereum still have possibility of replay-attack if transaction is to be made from the very old wallet (the wallet which have no associated transaction after the fork)?

No.

Two things are important:

1. The Ethereum foundation actually like the idea of forks, allowing different views in the community to co-exist. The Foundation only support one chain (ETH in this case).

2. Ethereum is a different type of Blockchain from Bitcoin at the moment. It contains contracts that could be automated to perform specific task. Which makes it easier to automate solutions for forks.

This post will be help explain how it was solved at the time https://blog.ethereum.org/2016/07/26/onward_from_the_hard_fork/