I'm trying to create an experiment involving the lac operon and for it I need to know a general timeline for protein synthesis. Is there a standard time in the bacteria's life where it begins production of the repressor protein for this operon? Where can I find this information?

In contrast to eukaryotic cells in most bacterial cells we do not distinguish between cell cycles, mostly because there are, with few exception, no easily traceable events that would make it useful. Perhaps I should specify that in older works (~70s-80s) it was more common to time bacterial cell cycle based on how far DNA replication progressed. But that is hardly being used anymore.

 

Rather, most characterization is done on the population level. That being said, the expression mostly depends on the promotor (obviously) which results in a constitutive expression. I.e. it is being expressed throughout the life-cycle of the cell. There is also evidence for a feedback inhibition which maintains it a low level.