How long can I run an Arudino circuit on a capacitor
I think your "back of the envelope" estimate is reasonable. 

For a series RC network with \$R=33\,\Omega\$ and \$C=1\,F\$, and initial voltage across the capacitor of \$V_C(t=0s)=10\,V\$, the voltage \$V_C\$ across the capacitor at any time \$t\ge0\,s\$ is given by:

$$
V_C(t) - V_F = \left( V_C(0)-V_F \right ) e^{-t/RC} \bigg\rvert_{V_F=0V}
$$

where \$V_F\$ is a forcing function term (e.g., a battery voltage), which in your case is not present, so you are only interested in the natural voltage decay response. The time \$t\$ when the voltage across the capacitor reaches 5.3&nbsp;Volts is

$$
\Rightarrow t = -RC \; ln \left ( \frac{V_C(t)}{V_C(0)} \right ) \bigg\rvert_{V_C(t)=5.3V,\;V_C(0)=10V}
\\
\Rightarrow t = -33\,ln(0.53)
\\
\Rightarrow t \approx 21 sec
$$

(edit)
My analysis ignores your statement that the 1&nbsp;F capacitor has "a rating of 10V". (To be honest, I missed that requirement when I first read your problem statement.) If you want your capacitor to have a long service life you should not routinely charge up the cap to its maximum rated voltage.