Walking is the best option as you can wander around the whole of Gili Air in about 90 minutes.
Bicycles are widely available for rent for Rp 30,000-50,000 per day. Some hotels might give you one as part of your accommodation package.
There are fewer horse carts cidomos here than on Gili Trawangan, but they are still not hard to find. If you are planning an early morning departure and have heavy bags, it is wise to ask your hotel the day before to book a cidomo for you. Short trips will cost about Rp 30,000 per person.