I need to make my faucet payment where a total of 0.001 BTC will be distributed among multiple recipients. I would like to know how much Tx fee should I pay for this ?
That depends on how many recipients you are paying, and how many inputs you are using to build your transaction.
The standard fee right now is 0.0001 BTC per kilobyte. Each input will be about 180 bytes or so, each output will be about 34 bytes, and there will be approximately 10 bytes in fixed size data in every transaction.