You want to know what value of capacitor(s) to put ahead and behind the 7809, but you haven't specified the sensor current and probably have never measured the Arduino operating current.
The answer to your 7809 capacitor question is
right on Fig 8. of the data sheet. Only the two capacitors shown (short leads, non-inductive) are required to suppress possible oscillation of the regulator. Making them bigger does nothing...
Of bigger concern is the
Vdrop spec for the 7809. If the input (output from the 12V supply sags below 11V, you are in trouble. You could go with a 7808.
The only other possibility to go fix the 12V supply by decreasing its ripple.