как будет правильнее? (>>)

По-моему, ни то и ни другоею
Old Riga, since the 13th century was the economic, financial and cultural center of the region, which eventually turned into a thriving city.

since?

Since 13th century Riga was the economic, financial and cultural center of the region etc.
Ну или бегининг виз я бы заменил на стартинг фром.

Old Riga, beginning with the 13 th century had become * the economic... center, which eventually turned into a thriving city

*Ну смысл такой, если уж потом ещё превратилась цветущий город, то тогда уж лучше использовать past perfect.