The Statue of Liberty can be visited at Christmas in Budapest: the statue is open all year round regardless of national holidays, including the days of Christmas. That said, Budapest public transport slows down slightly so you may need to plan your visit with less frequent buses, streetcars, etc. and in general a slower public transportation. If you don’t mind climbing, and are in an average shape regarding winter fitness, you can get to the top of Gellert Hill on the stairs, for instance from Gellert Baths, in about 15-25 min from the Liberty Bridge. Buses also go there, but less frequently. Do get dressed up well.
The robust lady with a palm tree is actually a Communist Liberation monument atop the Gellert Hill in Budapest, by the Citadel. There is a cafe and restaurant on the Citadel (its opening hours are not yet known during the Christmas holidays).
The Statue of Liberty is one of the very few Communist relics that managed to survive the change of regime in 1989.
Rather than pulled down and shoved to the Memento Statue Park, the collection of giga sized Communist statues, the Statue of Liberty somehow was accepted and embraced over time, despite the fact that all Hungarians find the Soviet liberation a controversial and grotesque issue: the Soviet troops liberated Hungary from the Nazis in 1945, but with the same gesture, the Soviets took over the control over Hungary, managing the actual political decision making, and rendering Hungarian politicians puppets.