To my knowledge, no such service exists, although building one would not be terribly difficult if you have access to a node.
Naturally, entrusting private keys to a third party is a terrible idea and you should absolutely not do this.
Using a smart contract is a lot easier. You can easily deploy a forwarding smart contract that forwards funds to a cold wallet, and simply share that contract's address with whoever is sending you ether. That way, regardless of when they send you money, it will automatically be forwarded to your address.
A very simple forwarding contract may look something like:
contract Forwarder {
address public destination;
constructor() public {
destination = msg.sender;
}
function() payable public {
destination.transfer(msg.value);
}
}