yes, besiktas pubs open until 4.00 am. Kadıkoy pubs open at 4.00 am. i am not visiting taksim late at night for years (nevizade street), but they are most problably will be open. i will ask them for you later. probably, since you are coming to town, taksim pubs will be open also. i will check.
it is forbidden to sell acohol at stores after 22.00. but some of the stores does. there is one in taksim center. there are others in taksim. also besiktas, kadıkoy, there are stores selling alcohol till 4.00 or something.
you are coming to town, and they know it, so they will be waiting for you :)
cheers.