Phototube experiment with op amp
You don't need a separate capacitor.
From the lab description you linked to:

The  physically  separated  anode  and  cathode  behave  as  half  of  a  cylindrical  capacitor;  the incoming  light  acts  as  a  current  source  to  charge  up  this  capacitor  (C)  with  electrons.    The total  charge  (Q)  and  potential  difference  on  this  capacitor  are  governed  by  a  fundamental equation  from  electrostatics:  Q  =  CV. 

The capacitor is an inherent part of the phototube.  Your task is to measure the accumulated charge between the plate and the rod of the phototube.
That will be a very small capacitance.  You will need an opamp with a very low leakage current on its input.
The charge current will be some few microamperes.  You'll need an opamp with a leakage much lower than that.
Your LMC6081 seems to meet that need - it has an input impedance in the teraohm range, and leakage in the picoampere range.
To make this work, you're going to need a setup designed for this kind of stuff.
I know I'm not up to it.  I have read a bit about this kind of thing (extreme high impedance measurements) and know that I don't have the skills needed.

At the impedances you are dealing with, a fingerprint on the PCB can act as a resistor and mess up your measurements.
Many things you would normally consider to be insulators begin to look like conductors in comparison with the impedances you are dealing with.
The input pins of the opamp will react to electric fields.  You will need to keep the connection to the phototube short, and shielded.
Look into guard rings, you're gonna need them.

This article from Analog Devices goes into some detail about things you will have to consider when building your circuit.

There are good reasons why the commercially available lab setups for this experiment in determining Planck's constant cost several hundred dollars.
A lot goes into them.  Keep going, and I expect you'll learn all about what makes them expensive.