First of all, a half day isn't really enough. Even if all you wanted to do was to walk around the shops of the three main streets in central Ubud and whizz through the markets, it would take you at least a few hours. If you're looking to book a driver for a half day, you're probably negotiating for 4 hours and it's going to take at least an hour each way to get there from the south.
The monkey forest is very central, but at the opposite end of Monkey Forest Road to the markets (which run along the main road through Ubud). The forest itself is small and pretty, as are the temples within it, but the monkeys can be intimidating. If you go, don't carry food.
If you're an early riser and you're interested in seeing the local market it could be worth booking the driver for a very early start (say 5am if you're coming from the south) and getting to the markets in time to see offerings and produce stalls before they pack up around 9/10am to make way for the souvenir stalls. That way you would miss the heavy traffic and the heat.
Taro Elephant Park is a 30 minute or so drive, north of Ubud central. If you were to book a mountain bike tour with Bali Adventure Tours on a separate day, you will finish in the elephant park (where you'll have lunch by the elephant pool and have time to feed them and interact with them before your car takes you back to your hotel).
Some other sites worth visiting near Ubud are Goa Gajah (the elephant cave), Gunung Kawi (burial complex with cliff carvings, reached via a long path through rice terraces), Gunung Kawi Sebatu (temple with many bathing pools for cleansing evil spirits), Tegallalang (rice terraces and lots of craft shops), Tegenungan waterfall.
Plus there is Bali Bird Park and Bali Zoo.