I'm not sure were you went wrong. In my first game I won a cultural victory with Alexander in the 1960ies. Had 4 cities and 3 puppets. Was pumping about 650 culture per turn. It's important to get all the -33% culture cost wonders and civics. I also allied all the cultural city states (about 4 I think) which were contributing 20 culture each. However I really concentrated on culture at the end, build all cultural improvements and had all culture specialists activated.