Your suggested transistor seems a bit on the weak side (for a full meter of ribbon) with 200 mA absolute maximum collector current. (Beyond that it may be destroyed) Instead I'd opt for the 4 times stronger BC337: https://www.sparkfun.com/products/13689
Make sure you check the emitter, base and collector pin out. There is no standard order.
I don't understand why you think the transistor outputs 6 volt. What made you think that? A transistor does not output anything by itself. It depends on the voltage applied across it. If you mean to use the 6 volt through 2 3 volt coincells then yes. But ... (the following)
I think you need to learn a bit more how to use transistors. https://learn.sparkfun.com/tutorials/transistors
What you described as how you would connect it is called common base configuration
That does not work as a switch. Instead you want to use the common emitter configuration. Base is connected to the micro:bit pin (through a current limiting resistor, 1000 Ohm should do) used to activate the ribbon. Emitter is connected to both (+ 4.5v) battery minus and ground of the micro:bit. Collector is connected to the minus of the led-ribbon. The plus of the led ribbon is connected to your +4.5 battery positive lead, making a full circuit. This circuit is described in Sparkfun's Transistor tutorial under the "Application 1" section.
You make a good point about a current limiting resistors for the led ribbon. I can't figure from the datasheet if the ribbon already contains them. Each led probably has one because that is just safe standard practice. And for even white/blue ledsabout 4 volts should be sufficient. The remaining 0.5 volt is probably dropped across a dedicated resistor at each led. But I would want to visually confirm it instead of assume. Your calculation is correct for 200mA. But that depends on the actual length of the ribbon.