Add a rechargeable LiPo to Arduino micro project
OLD ANSWER: There does not appear to be a USB pass through in that circuit, so you would not be able to download to the Arduino while you are charging (it's not clear whether you intend for the 'same usb cable' to mean the 'same usb cable at the same time'). If you want to design your own solution, there are a lot of simple USB based single cell lipo battery charging ICs (MAX1551, MCP73811...many others). 

EDIT: I didn't realize the Arduino micro is a 5V device, the part linked would not work. A single lipo cell only provides up to 4.2V (but realistically more like 3.9V under load), so there is no linear regulator which can provide the voltage the Arduino needs. So either you need to step up the voltage from the battery with a switching regulator, or you need a 2 cell battery/charger (which would need a step up anyway to be able to charge from USB). A simple solution would be to use https://www.sparkfun.com/products/11231, which integrates both the charge controller and a step up regulator. It also does not pass through USB, so you can't charge and program at the same time either...