Pin 1 and 3 (base and emitter) connected via 10k resistor and this should work awesomely yes?
Yes
did you do the math yourself on the need for these resistors or did you know this from experience?
The values are not too critical so while I have done the maths hundreds of times they always come out the same so you could say it is from experience.
The 10K is to ensure the transistor's base is not floating during power off of the arduino, anything from 10K to 100K will work just the same. The 1K is to ensure that the base current is limited to 4.2mA because a base voltage of 0.7V being driven from 5V will give 4.2V across the resistor giving 4.2mA base current. Then this base current is multiplied by the gain of the transistor to give the maximum load current. So suppose you had a gain of 100 then the maximum the load could be would be 420mA. As you did not give any figures for motor current I assumed that this would be sufficient. Keeping the current out of the arduino pin below 35mA is important for the health of the arduino, but apart from that it is all very loosely defined, as long as you are in the right ball park it will be fine.