A hill with a small joke on top
Petřín Lookout Tower, built for the 1891 Jubilee Exhibition, is a deliberate scaled-down homage to the Eiffel Tower, built two years after the Paris original. Climb it (or take the lift) and you get a rooftop view over Prague that rivals the castle's, minus the castle's crowds.
The funicular
Rather than walking up, most visitors take the Petřín funicular, a short cable railway that's been running (in various forms) since 1891 and is covered by the standard public transport ticket. It's a scenic ride in its own right, climbing through orchards and gardens on the way up.
What else is up there
Near the tower, the Mirror Maze is a small, genuinely fun distraction, especially with kids. The Hunger Wall, a fortification wall built in the 14th century, runs along part of the hill — according to local legend, built as a make-work project to feed the poor during a famine, though historians treat that story with some scepticism. In late spring, the hill's rose garden is worth the detour on its own.
Best time to go
Late afternoon into sunset. The light over the castle and river from the tower's viewing platform at that hour is one of the best free views in Prague, and the walk down through the gardens afterward is a good way to end a day on the castle side of the river.