It is not possible to send money anonymously using western union because upon receiving the money, the receiver needs to verify five informations on the money transfer; amount, the sender's name on record, receiver's name, originating city and state, city and state of payout.
If you are thinking about using another name, that is also not possible since western union agents are required to check the sender's ID in compliance with the anti-money laundering act. Let's say the agent forgot to check for your ID, it is still not good to use a different name because in case you would need to get a refund, you will be asked for the Original Receipt and your valid photo ID.
You can send the money via Western Union Overnight Home Delivery but you can only do that if you are sending the money via Debit or Credit Card over the phone by calling 1-800-225-5227. For first time users, you need to undergo the validation process. They will send a Western Union Debit card to the receiver's address and the payee calls a certain number to load up the card and the pin will be the control number from you. For further informations about that call their customer service at 1-800-325-6000.