Can you experimentally determine activation energy if the rate expression and rate constant are unknown?
This is valid only provided that the reaction is strictly first-order and provided that the “specific end-point” is a fixed concentration ratio. Note that the slope of $\ln(t_\text{end})$ vs. $\frac{1}{T}$ is $\frac{E_a}{R}$ but the intercept is not $\ln(A)$.

If the rection is first order ( $X {\xrightarrow{k}} Y$) then the product concentration at time t is:

$[Y]_t = [X]_0 .e^{k. t}$

ie:

$\ln\left( \frac{[Y]_t}{[X]_0}\right) = k. t $

If the end point is some fixed concentration ratio ($\frac{[Y]_{end}}{[X]_0}$), the time to reach the endpoint is:

$ t_{end} = \frac{\ln\left( \frac{[Y]_{end}}{[X]_0}\right)}{k}$

If we substitute this into the Arrhenius equation ($\ln(k) = \ln(A) - \frac{E_{a}}{R.T}$), we get:

$\ln(t_{end}) = \ln\left( \ln\left( \frac{[Y]_{end}}{[X]_0}\right)\right) - \ln A + \frac{E_{a}}{R.T} $

so a plot of $ln(t_{end})$ vs. $\frac{1}{T}$ will have a slope of $\frac{E_{a}}{R}$