With a rechargeable lithium ion batteries 9v, Can I connect it directly to the arduino pro mini? Or do I have to use a voltage regulator?
According to SparkFun’s site the two types of pro-minis work up to 12v on the raw input.
Here’s the datasheet for the onboard regulators: http://www.micrel.com/_PDF/mic5205.pdf
Remember to take thermal dissipation into consideration… Or it may get rather toasty. 
it depends on how much stuff will be powered through the regulator on the pro mini.
Im running a pro mini with 12V into RAW - powering the pro mini and a couple other ics that also use the pro mini regulator. The regulator is warm and i keep an eye on it. Im using the pro mini to send out logic to a bunch of other components that are directly on the 12v line. i think i would rethink this setup if i added even one LED to the mix. There are some nice switching 5V regulators available.
it all seems to be working fine.