We visited the city walls on our first day in Dubrovnik. You can get tickets just through the main gate of the old town for 10EUR. There are many steps to climb. Once you are at the top, you have a great view of the inside and the outside of the old town so be sure to bring a camera.
When we were there, there were a lot of other people also climbing the walls. Passages were narrow and we had to wait to pass at several points. We were told this is common in July and August.
The climb up the walls was worth every step and worth the 10 EUR. The views are incredible. Would recommend.