The general process of trying to solve this problem:
- Determine starting molarities of each reactant. Let's say 0.1 M
- Have a set of sample collection tubes (test tubes) pre-measured with 0.1 M of standardized HCl that would yield a slight excess of moles of NaOH in the starting mixture. This will be used to STOP the reaction (more on this). I would choose at least 5.
- Setup your reaction batch and mix the reactants with enough volume. Probably 100mL of each would suffice. As well as make sure the reactants and mixture maintain a temperature desired.
- During set intervals, withdrawal 10mL of the reaction mixture at periodic intervals and place them into one of the sample tubes. Placing it in the tube will immediately quench the NaOH in the reaction sample and convert any remaining acetate ions to acetic acid.
- After each sample tube has been used, you can titrate the remaining HCl + acetic acid with a standardized NaOH solution and an appropriate indicator such as bromthymol blue to obtain the amount of acid present.
You can now determine the amount of NaOH extracted from the reaction by using the number of moles of HCl present in the tube to start subtracted by the number of moles of acid that remain (step 5) to determine the amount of NaOH.
Since we now know the amount of NaOH extracted in a 10 mL sample of the reaction batch, you can determine the concentration of the NaOH at time of extraction within the reaction batch AND the time it took to reach this concentration.
Repeating this for each tube, we now have the change of the molarity of NaOH over time. This is also allows us to plot the change of the molarity of ethyl acetate since their reaction is a 1 mol : 1 mol ratio.
Using the integrated rate laws, a plot of these concentrations over time, you can determine the slope of the plot which creates a linear graph and thus find the reaction constant $(k)$.
Thanks to @Poutnik for the help on refining the answer.