Ice skating
Open-air skating rinks: only during winter
You can do several things in Budapest: caving under the city, paintball, gokarting, kayaking / canoeing on river Danube.
City Park's lake
bicycling
Unfortunately, there are few bicycle lanes and many dangerous drivers. However with organized groups such as Critical Mass (http://criticalmass.hu/) operating in the city, things are slowly becoming more bicycle friendly. Pest is very flat and excellent for bicycle riding - just make sure to keep your wits about you, as drivers can be somewhat careless.
There are several places in Pest to rent bicycles:
TreeHugger Dans (http://www.treehugger.hu/) - Used english bookstore, bicycle rental and coffee shop. Smoke free and the staff is very friendly even if the book prices are extortionate. A good place to pick up a local city map with fresh recommendations.
Yellow Zebra Bikes (http://www.yellowzebrabik...) Rents bicycle and also provides bicycle tours of the city