In Ohio, the sun will set before the planet Venus reaches the midpoint of its journey across the face of the sun. Venus will finish its transit by completely leaving the face of the sun at 12:50am on June 6, obviously well after the sun is out of sight.
There is not a lot of difference in times between cities in Ohio. Cities in the east will see the transit begin only seconds before cities closer to the Indiana border. Below is a map that includes the external ingress times for major cities, as calculated by NASA.
The transit will not be visible everywhere in the world, so we are lucky to see several hours of the transit before sunset! Here is a world map, provided by NASA (PDF), that shows the regions where the transit will be visible.
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