Nth-Light LED Strip power connections
If your 12V power supply has enough current to drive your LED strip at its maximum power, plus the Arduino's power supply, plus about 10-20% of "wiggle room" you should be fine. As the LED strip draws spikes of current the voltage from your power supply might bounce around a little, but the Arduino's voltage regulator would be fine with as little as 7.5V input.

Note that your Arduino's voltage regulator will get hot if you use much current from the Ardunio's 5V rails, since it has to regulate 12V down to 5V and the linear regulator on the Arduino converts the extra voltage to heat.