Currently TripAdvisor has over 1100 reviews of the Dubrovnik city walls but only 24 of Fort Lovrijenac -- a clear indication of the relative popularity. Yet it's absolutely worth your while to make the climb to the fort after seeing the walls. See the walls first to avoid the crowds there, the fort is usually quiet in any case (we only saw four other tourists there during our visit). Or visit the fort at sunset for some great photos.
The fort is only 15 minutes' walk from the Pile Gate. You need to walk down the first street outside the gate and then navigate some other streets while keeping an eye on the fort; it's not difficult to find but the lack of street signs makes it a little bit less than obvious.
Once you've descended to the little bay below Pile Gate, you will see the path climbing up to the fort through a small park-like area. The climb is not bad if you are reasonably fit, and already offers some views.
Once inside the fort, you will need to pay admission (currently HRK 30) unless you have already paid for the city walls. If you have, that ticket will get you free admission to the fort. You can then climb up several levels of stairs to the top of the fort. The main attraction will be the views of the old city, and the fort's distance from the walls allows you to capture an overview of Dubrovnik, which is more difficult to do from the wall itself.
Don't miss it!
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.