> What kind of capacitor and size would you consider suitable?

I'm of the suck-it-and-see school of components so maybe
I haven't got this quite right. I'm happy to be corrected so I
may learn

Your circuit, using 25mA at 3V3 => 3.3/0.025A = 132ohms

Say you have a 100uF capacitor at 5.0V. The RC time constant
is 132R * 0.0001F = 1.32ms, 5 times which is what you would
allow to get the voltage up

Charge (Q) is C*V = 0.0001 * 5.0 = 500 uCoulombs

Assuming it takes a generous 50us to do the ADC, in 50us
the capacitor will have lost

0.025A * 0.000050s = 1.25uC

=> charge is now 498.75uC or 99.75% or 4.9875V

If the initial charge was to a little over 5.12V, the actual voltage
would have dropped to 5.12V at the time of acquisition, making
calcs a little easier and no fractions of bits to help results

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