I am aware that it already has a regulator, but should I be worrying about how clean the power is going to the board?
With 12V going-in and the Arduino only needing 5V, you are giving the regulator plenty of "room to work" and the voltage regulator should hold the voltage steady. So, the on-board regulator should be fine. If there's any noise or glitches getting through, a 2nd or different regulator is unlikely to help.
If you do run into problems, such as the voltage dropping too-low for a moment when you hit the starter, or when you first turn-on the headlight, etc., you can try adding a diode and capacitor (maybe 1000uF or so) to the 12V input. This makes sort-of a "one-way" filter... It won't do much for positive
spikes on the power-supply line (the regulator should take care of that), but if there is a negative
-spike where the voltage momentarily drops below 5 or 6 volts, the capacitor will hold-up the voltage (hopefully long-enough) and the diode makes sure all of the energy stored in the capacitor goes to the Arduino, by prevening the capacitor from quickly-discharging into the overall electrical system, and "trying" to power everything else on the scooter.
And actually... as long as the thing resets properly when the voltage drops-out, and it doesn't shut-down the scooter when there is an unexpected reset, it's probably not too critical .