Why does Green Tea taste sour as it cools down?
The most likely culprits are:


Steeping too long: 1-3 minutes, depending on the type and grade of tea.
Using water that is too hot or too cold: Again, this depends on the type and grade of tea.


Here is an excellent resource for getting these factors right:

On Steeping Time:


  Green tea does not require much time. Too long a steeping time will result in more bitterness and a less balanced flavor. We recommend experimenting with a range of 1 - 3 minutes. Japanese green teas generally taste best at 1 - 2 minutes while Chinese green teas seem to prefer 2 - 3 minutes (the smaller leaves of Japanese teas will extract faster than the generally larger leaves of Chinese teas). Steeping time should be balanced with water temperature: the lower the temperature, the longer the tea can be steeped.


On Temperature


  As a general guideline, green teas taste best when brewed at temperatures between 140°F - 185°F (60°C-85°C). The grade of the tea and the time of its harvest will also influence the appropriate steeping temperature. Green teas picked earlier in the spring will benefit from lower temperature brewing due to their overall higher levels of amino acids.


...


  The most accurate way is to use a thermometer to measure the temperature of the water in the kettle. One approach is to heat the water in your kettle to the desired temperature and then pour it into your teapot. Another approach is to heat the water to boiling and then let it cool down a bit before pouring into your teapot.


Read the whole thing for further background.