Old Wallet, Private key and known public address This is the scenario 
I have an old wallet 
This wallet misses a transaction. This transaction was confirmed several times in 2014
I have the public address of that transaction 
If I ask to my friend to send the payment again (how?) to my address , will I get my founds back again?
In other words, I have an old backup of a wallet that misses a transaction but I have the pubblic address where my friend sent the coins
Is thera a way to get the litecoin back again without knowing the private key?Consider that I have the original wallet
Your wallet will contain a collection of private keys, and it should show you the balance of the addresses associated with those keys.

For the public address you mentioned, if you have the private key associated with it, you'll be able to spend the coins. If not, there is no way to recover a private key from a public key. The private key is either known by the wallet file, or not. There is no way to 'resend' a payment, etc, to recover a private key that is otherwise not known.