Lipo/Bluetooth and Arduino

I'm trying to create a device to wirelessly control my desktop (input key presses). I was thinking something rechargeable like a lipo and bluetooth for the actual wireless communication (saves the hassle of making a receiver if using something like IR or RF). I found this:

Could I use this to power the arduino? Just the microcontroller, not the whole board since it will be a 3.7v lipo. Would I use the terminals for the battery to also power the microcontroller? And using the bluetooth would be just hooking up the Tx/Rx pins between them?