Abu Dhabi is a great place, in fact I'm going to live there in 2 months time.
If you're only there for 24 hours, I'd recommend:
1) Shopping at the Marina Mall in the morning/early afternoon.
2) Taking a walk through Khalifa Park in late afternoon.
3) Taking a stroll along the corniche in the evening as the sun is setting (beautiful views).
4) Using the hotel's private beach/swimming pool in the late evening when it's cooled down.
Don't bother walking between places, it's too hot, the roads are confusing for pedestrians and it will take ages. Taxis are very cheap and will cost $1.50 to cross the entire city. You can just flag one down, however there's normally a few outside each hotel.
Be prepared to show respect when in public, long sleeved shirts are sensible and if wearing shorts, make sure they come just below the knee. Also, the coins don't have any "normal numbers" like 50 on them, so you might want to learn the Arabic numerals before going!
If you have a German passport, then you're correct, you won't need a visa. Just be prepared to answer "have you been here before?" - they always ask this at the immigration counter in AUH
. It's no pain at all and they're very friendly.