Suggest a method, other than spectrophotometry, for determining the reaction rate
Spectroscopy is often the best method, since it allows for (almost) continuous monitoring of the concentration.

If that is not wanted, ask yourself: 


What is released in the hydrolysis of the complex?


Chloride would be a good guess.


Can I take samples and perform a wet chemical analysis to determine the concentration of new species?


Looking at the graph in your other related question, the answer is yes. The hydrolysis is slow enough.


Which methods can I possibly use to determine the concentration $\ce{Cl-}$ in solution?


An argentometric titration, based on the low solubility of $\ce{AgCl}$ is a good idea. What about the Mohr titration, using $\ce{K2CrO4}$ as an indicator?