The regulator on the Arduino is a linear regulator, which means that it reduces the voltage by throwing the rest of the power away. At 9V and 650mA, it's throwing away (9V - 5V) * 650mA = 2.6 watts. This is a decent amount of power, and more than the regulator can handle.
Using a switching regulator instead would cause it to make up for the drop in voltage by using less current from the source; a 90% efficient switching regulator would waste only about 5V * 650mA * 10% = 325mW, which is easier for larger packages to disperse.
Look on eBay or DX or the like for a 5V or adjustable DC-DC switching module. Connect the input to your 9V source and the output (set to 5V) to both the 5V input on the Arduino and the 5V input on the LED strip.
Interpreting specs such as these found on many marketplace sites can be a bit of an art form, especially when no datasheets are provided.
When in doubt, the best idea is to ask the seller. If they cannot provide the information, then you have two choices: take pot luck, or find another seller!
That being said, here's how I read the information you are being provided in this case:
- Input Voltage: Dc 12V – 3A
.. is I believe the spec of the power supply that is provided, not the strip itself. In other words, capable of delivering a total of 36W at 12V.
- Lamp power: 12V 4.8 Watts
.. is most likely per meter i.e. each 60 LEDs. That is a pure guess based typical power requirements and how specs for similar LED strips are presented. You can divide/multiply this out by the actual number of LEDs you plan to use.
If you are planning to power a strip with batteries, the main challenges will be battery life and how many LEDs you have the power to drive.
Trying to use 2x9V does not seem like a very good idea. While you could probably drive at least a few LEDs with 2x9V, you will be wasting a lot of their already-limited (~500mAh) capacity in the step-down to 12V. And a step-down with resistors is going to be tough to get right. If you cut the strip and can squeeze say 0.5A out of the batteries, that's still 3W to burn off. A voltage regulator would be better if you really must.
AA or even C, D or 6V lantern batteries will be able to drive more LEDs for longer, and you can avoid losses in voltage regulation.
Best Answer
90W at 5V comes out to 18A. You can find several 5V 20A power supplies around.
P(power)=V(volts)*I(amps) therefore I=P/V 90W/5V=18A
I have built several similar projects using WS2812b with Arduino and LPD8806 with Raspberry Pi. I used this 5V 60A power supply and it has been running in my attic for years with no issues. Its a bit large for what you need but at $20, you're not going to find much cheaper.
Unrelated to the power supply, its a lot easier to use the LPD8806 or some similar 4 pin cable with RPi because you can control the clock. The three pin addressable strips need to have their data output at a very specific frequency and it can be troublesome to run long strips. It can be done, especially with the short run you have, and the 3 pin version are usually cheaper.
P.S. Hardware Recommendations is a good site for these kinds of questions.