How much time for a water molecule in pure water to reach the top of a glass of water?
I am posting here the answer since it has been quickly found thanks to the useful comments. To give an estimate the best way is to look at the self-diffusion coefficient.
This is defined as the diffusion coefficient when the chemical potential gradient equals zero, or more simply, it is the diffusion coefficient in a medium of the same molecules.
According to Ref.1, the self-diffusion coefficient of water at $\pu{25^\circ C}$ is $D = \pu{2.3 \times 10^{−9} m^2 s^{−1}}$.  The diffusion time can be calculated as: $$t \approx \frac{x^2}{2D}   $$
As pointed out by @Karsten Theis, you can use the calculator and insert the water diffusion coefficient D.
For example, if the distance x is $\pu{8cm}$ like a typical glass of water and we are at $\pu{25 ^\circ C}$, then $t \approx \pu{16 days}$!
Instead if the temperature is $\pu{90 ^\circ C}$ then it's around $\pu{5 days}$.
I have drawn a graph that shows how much time to reach the top of the glass depending on temperature:

References:

Manfred Holz, Stefan R. Heila, Antonio Sacco, &quot;Temperature-dependent self-diffusion coefficients of water and six selected molecular liquids for calibration in accurate $\ce{^1H}$ NMRPFG measurements,&quot; Phys. Chem. Chem. Phys. 2000, 2(20), 4740-4742 (https://doi.org/10.1039/B005319H).