In Y-town all crossroads are Y-shaped, and there are no dead-end roads. Is it true that if you start from any point in the city and start walking along the roads, turning alternatingly left and right at each crossroad, eventually you will arrive at the same spot?

Yes, it is true. If you start walking, eventually your path will form a repeating cycle. Assume your starting position does not belong to that cycle. Now it is easy to see that if you start walking backwards, your path once again will form a cycle. However, this will imply that you could not have started from your initial position (since it is not part of the cycle), which is a contradiction.