Motorcycle Battery Powering Arduino UNO
Your circuit is 'going in the right direction' but can be improved.
If you wish to use the regulator shown:

The larger the input capacitor (C1) the better; it will survive noise spikes and low voltage &quot;droops&quot;, eg during starting.

Adding a series resistor to drop some of the input voltage will help C1 filter out noise.
Rin should be V/I = V_drop_max/I_maximum .
With a 9V regulator and 12V supply you cannot afford much drop in the resistor.
If you use the 6V output system I recommend below then you could allow closer to 3V drop in Rin.
If Imax is 100mA then Rin = V/I = 3/0.1 = 30 Ohms.
Typically you'd use 27 Ohms (which is the closest E12 standard value.


Power dissipation is the resistor = I^2 x R.
So for eg 27 Ohms and 0.1 A (100 mA) power dissipation in the resistor is
I^2 x R = 0.1^2 x 27 = 0.27 Watt
So you'd use at least a 0.5 Watt resistor and ideally  Watt or more. Air cooled resistors are relatively low cost and a much better way of dealing with heat than trying to cool a regulator.

Adding a ~15V zener diode at the input to U1 will help protect against spikes. This works better if Rin is used. Without Rin the zener may be destroyed by spikes. If you use the LM29xx regulators that I mention below the zener is not needed as the regulator has its own internal protection circuitry.

The LM7809 regulator is somewhat unusual. If you have them available they could be used but there are much more common and/or modern regulators available.
The regulator's Vout should be fed to the Arduino's Vin terminal, which supplies the Arduino's internal regulator, to provide 5V or 3V3 depending on which Arduino you are using. With a 5V Arduino, using Vin of 6V or more will be adequate. Higher Vin causes more heat in the Arduino on board regulator. (As Chaaarlie2 noted, 6V on Vin may be a bit on the low side for heavy loads on the Arduino 5V supply - as may happen if shields use the 5V supply. Increasing Vin to say 6.5V should allow the Arduino's internal regulator to supply its maximum rated current if desired.
9V output is higher than the Arduino needs. It is an acceptable voltage but 6V to 7V will cause less heat on-board the Arduino (and more in the external regulator).
Vehicle power systems can have extremely high voltage spikes present and sudden changes in supply voltage. Some regulators are made to work well in the automotive environment. One such family are the LM29xx series of regulators.
One superb version of this regulator is the LM2931-N [Data sheet here] - this is available in a range of versions but the simplest is the 3 lead type in a range of packages. The TO220 package is recommended due to its good power dissipation capability and ease of heatsinking if necessary. This data sheet covers versions with about 100 mA output rating, but higher output examples are available. Fixed 3.3V and 5V output versions are available but in this application the variable output version, set to about 6V or maybe 6.5V is &quot;safest&quot;. These are very similar to the common LM317 in operation. The basic regulator produces 1.26V output and 2 resistors are used to set the desired output voltage. The advantages over the LM317 are the lower dropout voltage and the protection against typical automotive input problems.
The datasheet description states:

Designed originally for automotive applications, the LM2931-N and all regulated circuitry are protected from reverse battery installations or 2 battery jumps.
During line transients, such as a load dump (60V) when the input voltage to the regulator can momentarily exceed the specified maximum operating voltage, the regulator will automatically shut down to protect both internal circuits and the load.
The LM2931-N cannot be harmed by temporary mirror-image insertion. Familiar regulator features such as short circuit and thermal overload protection are also provided.